Miscellaneous arithmetic functions

sage.arith.misc.CRT(a, b, m=None, n=None)

Return a solution to a Chinese Remainder Theorem problem.

INPUT:

• a, b - two residues (elements of some ring for which extended gcd is available), or two lists, one of residues and one of moduli.
• m, n - (default: None) two moduli, or None.

OUTPUT:

If m, n are not None, returns a solution \(x\) to the simultaneous congruences \(x\equiv a \bmod m\) and \(x\equiv b \bmod n\), if one exists. By the Chinese Remainder Theorem, a solution to the simultaneous congruences exists if and only if \(a\equiv b\pmod{\gcd(m,n)}\). The solution \(x\) is only well-defined modulo \(\text{lcm}(m,n)\).

If a and b are lists, returns a simultaneous solution to the congruences \(x\equiv a_i\pmod{b_i}\), if one exists.

See also

• CRT_list()

EXAMPLES:

Using crt by giving it pairs of residues and moduli:

sage: crt(2, 1, 3, 5)
11
sage: crt(13, 20, 100, 301)
28013
sage: crt([2, 1], [3, 5])
11
sage: crt([13, 20], [100, 301])
28013

You can also use upper case:

sage: c = CRT(2,3, 3, 5); c
8
sage: c % 3 == 2
True
sage: c % 5 == 3
True

Note that this also works for polynomial rings:

sage: K.<a> = NumberField(x^3 - 7)
sage: R.<y> = K[]
sage: f = y^2 + 3
sage: g = y^3 - 5
sage: CRT(1,3,f,g)
-3/26*y^4 + 5/26*y^3 + 15/26*y + 53/26
sage: CRT(1,a,f,g)
(-3/52*a + 3/52)*y^4 + (5/52*a - 5/52)*y^3 + (15/52*a - 15/52)*y + 27/52*a + 25/52

You can also do this for any number of moduli:

sage: K.<a> = NumberField(x^3 - 7)
sage: R.<x> = K[]
sage: CRT([], [])
0
sage: CRT([a], [x])
a
sage: f = x^2 + 3
sage: g = x^3 - 5
sage: h = x^5 + x^2 - 9
sage: k = CRT([1, a, 3], [f, g, h]); k
(127/26988*a - 5807/386828)*x^9 + (45/8996*a - 33677/1160484)*x^8 + (2/173*a - 6/173)*x^7 + (133/6747*a - 5373/96707)*x^6 + (-6/2249*a + 18584/290121)*x^5 + (-277/8996*a + 38847/386828)*x^4 + (-135/4498*a + 42673/193414)*x^3 + (-1005/8996*a + 470245/1160484)*x^2 + (-1215/8996*a + 141165/386828)*x + 621/8996*a + 836445/386828
sage: k.mod(f)
1
sage: k.mod(g)
a
sage: k.mod(h)
3

If the moduli are not coprime, a solution may not exist:

sage: crt(4,8,8,12)
20
sage: crt(4,6,8,12)
Traceback (most recent call last):
...
ValueError: No solution to crt problem since gcd(8,12) does not divide 4-6

sage: x = polygen(QQ)
sage: crt(2,3,x-1,x+1)
-1/2*x + 5/2
sage: crt(2,x,x^2-1,x^2+1)
-1/2*x^3 + x^2 + 1/2*x + 1
sage: crt(2,x,x^2-1,x^3-1)
Traceback (most recent call last):
...
ValueError: No solution to crt problem since gcd(x^2 - 1,x^3 - 1) does not divide 2-x

sage: crt(int(2), int(3), int(7), int(11))
58

crt also work with numpy and gmpy2 numbers:

sage: import numpy
sage: crt(numpy.int8(2), numpy.int8(3), numpy.int8(7), numpy.int8(11))
58
sage: from gmpy2 import mpz
sage: crt(mpz(2), mpz(3), mpz(7), mpz(11))
58
sage: crt(mpz(2), 3, mpz(7), numpy.int8(11))
58

sage.arith.misc.CRT_basis(moduli)

Return a CRT basis for the given moduli.

INPUT:

• moduli - list of pairwise coprime moduli \(m\) which admit an

  extended Euclidean algorithm

OUTPUT:

• a list of elements \(a_i\) of the same length as \(m\) such that \(a_i\) is congruent to 1 modulo \(m_i\) and to 0 modulo \(m_j\) for \(j\not=i\).

Note

The pairwise coprimality of the input is not checked.

EXAMPLES:

sage: a1 = ZZ(mod(42,5))
sage: a2 = ZZ(mod(42,13))
sage: c1,c2 = CRT_basis([5,13])
sage: mod(a1*c1+a2*c2,5*13)
42

A polynomial example:

sage: x=polygen(QQ)
sage: mods = [x,x^2+1,2*x-3]
sage: b = CRT_basis(mods)
sage: b
[-2/3*x^3 + x^2 - 2/3*x + 1, 6/13*x^3 - x^2 + 6/13*x, 8/39*x^3 + 8/39*x]
sage: [[bi % mj for mj in mods] for bi in b]
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]

sage.arith.misc.CRT_list(v, moduli)

Given a list v of elements and a list of corresponding moduli, find a single element that reduces to each element of v modulo the corresponding moduli.

See also

• crt()

EXAMPLES:

sage: CRT_list([2,3,2], [3,5,7])
23
sage: x = polygen(QQ)
sage: c = CRT_list([3], [x]); c
3
sage: c.parent()
Univariate Polynomial Ring in x over Rational Field

It also works if the moduli are not coprime:

sage: CRT_list([32,2,2],[60,90,150])
452

But with non coprime moduli there is not always a solution:

sage: CRT_list([32,2,1],[60,90,150])
Traceback (most recent call last):
...
ValueError: No solution to crt problem since gcd(180,150) does not divide 92-1

The arguments must be lists:

sage: CRT_list([1,2,3],"not a list")
Traceback (most recent call last):
...
ValueError: Arguments to CRT_list should be lists
sage: CRT_list("not a list",[2,3])
Traceback (most recent call last):
...
ValueError: Arguments to CRT_list should be lists

The list of moduli must have the same length as the list of elements:

sage: CRT_list([1,2,3],[2,3,5])
23
sage: CRT_list([1,2,3],[2,3])
Traceback (most recent call last):
...
ValueError: Arguments to CRT_list should be lists of the same length
sage: CRT_list([1,2,3],[2,3,5,7])
Traceback (most recent call last):
...
ValueError: Arguments to CRT_list should be lists of the same length

sage.arith.misc.CRT_vectors(X, moduli)

Vector form of the Chinese Remainder Theorem: given a list of integer vectors \(v_i\) and a list of coprime moduli \(m_i\), find a vector \(w\) such that \(w = v_i \pmod m_i\) for all \(i\). This is more efficient than applying CRT() to each entry.

INPUT:

• X - list or tuple, consisting of lists/tuples/vectors/etc of integers of the same length
• moduli - list of len(X) moduli

OUTPUT:

• list - application of CRT componentwise.

EXAMPLES:

sage: CRT_vectors([[3,5,7],[3,5,11]], [2,3])
[3, 5, 5]

sage: CRT_vectors([vector(ZZ, [2,3,1]), Sequence([1,7,8],ZZ)], [8,9])
[10, 43, 17]

class sage.arith.misc.Euler_Phi

Return the value of the Euler phi function on the integer n. We defined this to be the number of positive integers <= n that are relatively prime to n. Thus if n<=0 then euler_phi(n) is defined and equals 0.

INPUT:

• n - an integer

EXAMPLES:

sage: euler_phi(1)
1
sage: euler_phi(2)
1
sage: euler_phi(3)
2
sage: euler_phi(12)
4
sage: euler_phi(37)
36

Notice that euler_phi is defined to be 0 on negative numbers and 0.

sage: euler_phi(-1)
0
sage: euler_phi(0)
0
sage: type(euler_phi(0))
<type 'sage.rings.integer.Integer'>

We verify directly that the phi function is correct for 21.

sage: euler_phi(21)
12
sage: [i for i in range(21) if gcd(21,i) == 1]
[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]

The length of the list of integers ‘i’ in range(n) such that the gcd(i,n) == 1 equals euler_phi(n).

sage: len([i for i in range(21) if gcd(21,i) == 1]) == euler_phi(21)
True

The phi function also has a special plotting method.

sage: P = plot(euler_phi, -3, 71)

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: euler_phi(int8(37))
36
sage: from gmpy2 import mpz
sage: euler_phi(mpz(37))
36

AUTHORS:

• William Stein
• Alex Clemesha (2006-01-10): some examples

plot(xmin=1, xmax=50, pointsize=30, rgbcolor=(0, 0, 1), join=True, **kwds)

Plot the Euler phi function.

INPUT:

• xmin - default: 1
• xmax - default: 50
• pointsize - default: 30
• rgbcolor - default: (0,0,1)
• join - default: True; whether to join the points.
• **kwds - passed on

EXAMPLES:

sage: p = Euler_Phi().plot()
sage: p.ymax()
46.0

sage.arith.misc.GCD(a, b=None, **kwargs)

Return the greatest common divisor of a and b.

If a is a list and b is omitted, return instead the greatest common divisor of all elements of a.

INPUT:

• a,b – two elements of a ring with gcd or
• a – a list or tuple of elements of a ring with gcd

Additional keyword arguments are passed to the respectively called methods.

OUTPUT:

The given elements are first coerced into a common parent. Then, their greatest common divisor in that common parent is returned.

EXAMPLES:

sage: GCD(97,100)
1
sage: GCD(97*10^15, 19^20*97^2)
97
sage: GCD(2/3, 4/5)
2/15
sage: GCD([2,4,6,8])
2
sage: GCD(srange(0,10000,10))  # fast  !!
10

Note that to take the gcd of \(n\) elements for \(n \not= 2\) you must put the elements into a list by enclosing them in [..]. Before trac ticket #4988 the following wrongly returned 3 since the third parameter was just ignored:

sage: gcd(3, 6, 2)
Traceback (most recent call last):
...
TypeError: gcd() takes ...
sage: gcd([3, 6, 2])
1

Similarly, giving just one element (which is not a list) gives an error:

sage: gcd(3)
Traceback (most recent call last):
...
TypeError: 'sage.rings.integer.Integer' object is not iterable

By convention, the gcd of the empty list is (the integer) 0:

sage: gcd([])
0
sage: type(gcd([]))
<type 'sage.rings.integer.Integer'>

class sage.arith.misc.Moebius

Return the value of the Möbius function of abs(n), where n is an integer.

DEFINITION: \(\mu(n)\) is 0 if \(n\) is not square free, and otherwise equals \((-1)^r\), where \(n\) has \(r\) distinct prime factors.

For simplicity, if \(n=0\) we define \(\mu(n) = 0\).

IMPLEMENTATION: Factors or - for integers - uses the PARI C library.

INPUT:

• n - anything that can be factored.

OUTPUT: 0, 1, or -1

EXAMPLES:

sage: moebius(-5)
-1
sage: moebius(9)
0
sage: moebius(12)
0
sage: moebius(-35)
1
sage: moebius(-1)
1
sage: moebius(7)
-1

sage: moebius(0)   # potentially nonstandard!
0

The moebius function even makes sense for non-integer inputs.

sage: x = GF(7)['x'].0
sage: moebius(x+2)
-1

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: moebius(int8(-5))
-1
sage: from gmpy2 import mpz
sage: moebius(mpz(-5))
-1

plot(xmin=0, xmax=50, pointsize=30, rgbcolor=(0, 0, 1), join=True, **kwds)

Plot the Möbius function.

INPUT:

• xmin - default: 0
• xmax - default: 50
• pointsize - default: 30
• rgbcolor - default: (0,0,1)
• join - default: True; whether to join the points (very helpful in seeing their order).
• **kwds - passed on

EXAMPLES:

sage: p = Moebius().plot()
sage: p.ymax()
1.0

range(start, stop=None, step=None)

Return the Möbius function evaluated at the given range of values, i.e., the image of the list range(start, stop, step) under the Möbius function.

This is much faster than directly computing all these values with a list comprehension.

EXAMPLES:

sage: v = moebius.range(-10,10); v
[1, 0, 0, -1, 1, -1, 0, -1, -1, 1, 0, 1, -1, -1, 0, -1, 1, -1, 0, 0]
sage: v == [moebius(n) for n in range(-10,10)]
True
sage: v = moebius.range(-1000, 2000, 4)
sage: v == [moebius(n) for n in range(-1000,2000, 4)]
True

class sage.arith.misc.Sigma

Return the sum of the k-th powers of the divisors of n.

INPUT:

• n - integer
• k - integer (default: 1)

OUTPUT: integer

EXAMPLES:

sage: sigma(5)
6
sage: sigma(5,2)
26

The sigma function also has a special plotting method.

sage: P = plot(sigma, 1, 100)

This method also works with k-th powers.

sage: P = plot(sigma, 1, 100, k=2)

AUTHORS:

• William Stein: original implementation
• Craig Citro (2007-06-01): rewrote for huge speedup

plot(xmin=1, xmax=50, k=1, pointsize=30, rgbcolor=(0, 0, 1), join=True, **kwds)

Plot the sigma (sum of k-th powers of divisors) function.

INPUT:

• xmin - default: 1
• xmax - default: 50
• k - default: 1
• pointsize - default: 30
• rgbcolor - default: (0,0,1)
• join - default: True; whether to join the points.
• **kwds - passed on

EXAMPLES:

sage: p = Sigma().plot()
sage: p.ymax()
124.0

sage.arith.misc.XGCD(a, b)

Return a triple (g,s,t) such that \(g = s\cdot a+t\cdot b = \gcd(a,b)\).

Note

One exception is if \(a\) and \(b\) are not in a principal ideal domain (see Wikipedia article Principal_ideal_domain), e.g., they are both polynomials over the integers. Then this function can’t in general return (g,s,t) as above, since they need not exist. Instead, over the integers, we first multiply \(g\) by a divisor of the resultant of \(a/g\) and \(b/g\), up to sign.

INPUT:

• a, b - integers or more generally, element of a ring for which the xgcd make sense (e.g. a field or univariate polynomials).

OUTPUT:

• g, s, t - such that \(g = s\cdot a + t\cdot b\)

Note

There is no guarantee that the returned cofactors (s and t) are minimal.

EXAMPLES:

sage: xgcd(56, 44)
(4, 4, -5)
sage: 4*56 + (-5)*44
4

sage: g, a, b = xgcd(5/1, 7/1); g, a, b
(1, 3, -2)
sage: a*(5/1) + b*(7/1) == g
True

sage: x = polygen(QQ)
sage: xgcd(x^3 - 1, x^2 - 1)
(x - 1, 1, -x)

sage: K.<g> = NumberField(x^2-3)
sage: g.xgcd(g+2)
(1, 1/3*g, 0)

sage: R.<a,b> = K[]
sage: S.<y> = R.fraction_field()[]
sage: xgcd(y^2, a*y+b)
(1, a^2/b^2, ((-a)/b^2)*y + 1/b)
sage: xgcd((b+g)*y^2, (a+g)*y+b)
(1, (a^2 + (2*g)*a + 3)/(b^3 + (g)*b^2), ((-a + (-g))/b^2)*y + 1/b)

Here is an example of a xgcd for two polynomials over the integers, where the linear combination is not the gcd but the gcd multiplied by the resultant:

sage: R.<x> = ZZ[]
sage: gcd(2*x*(x-1), x^2)
x
sage: xgcd(2*x*(x-1), x^2)
(2*x, -1, 2)
sage: (2*(x-1)).resultant(x)
2

Tests with numpy and gmpy2 types:

sage: from numpy import int8
sage: xgcd(4,int8(8))
(4, 1, 0)
sage: xgcd(int8(4),int8(8))
(4, 1, 0)
sage: from gmpy2 import mpz
sage: xgcd(mpz(4), mpz(8))
(4, 1, 0)
sage: xgcd(4, mpz(8))
(4, 1, 0)

sage.arith.misc.algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False)

Return an irreducible polynomial of degree at most \(degree\) which is approximately satisfied by the number \(z\).

You can specify the number of known bits or digits of \(z\) with known_bits=k or known_digits=k. PARI is then told to compute the result using \(0.8k\) of these bits/digits. Or, you can specify the precision to use directly with use_bits=k or use_digits=k. If none of these are specified, then the precision is taken from the input value.

A height bound may be specified to indicate the maximum coefficient size of the returned polynomial; if a sufficiently small polynomial is not found, then None will be returned. If proof=True then the result is returned only if it can be proved correct (i.e. the only possible minimal polynomial satisfying the height bound, or no such polynomial exists). Otherwise a ValueError is raised indicating that higher precision is required.

ALGORITHM: Uses LLL for real/complex inputs, PARI C-library algdep command otherwise.

Note that algebraic_dependency is a synonym for algdep.

INPUT:

• z - real, complex, or \(p\)-adic number
• degree - an integer

• height_bound - an integer (default: None) specifying the maximum

  coefficient size for the returned polynomial

• proof - a boolean (default: False), requires height_bound to be set

EXAMPLES:

sage: algdep(1.888888888888888, 1)
9*x - 17
sage: algdep(0.12121212121212,1)
33*x - 4
sage: algdep(sqrt(2),2)
x^2 - 2

This example involves a complex number:

sage: z = (1/2)*(1 + RDF(sqrt(3)) *CC.0); z
0.500000000000000 + 0.866025403784439*I
sage: algdep(z, 6)
x^2 - x + 1

This example involves a \(p\)-adic number:

sage: K = Qp(3, print_mode = 'series')
sage: a = K(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: algdep(a, 1)
19*x - 7

These examples show the importance of proper precision control. We compute a 200-bit approximation to \(sqrt(2)\) which is wrong in the 33’rd bit:

sage: z = sqrt(RealField(200)(2)) + (1/2)^33
sage: p = algdep(z, 4); p
227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088
sage: factor(p)
227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088
sage: algdep(z, 4, known_bits=32)
x^2 - 2
sage: algdep(z, 4, known_digits=10)
x^2 - 2
sage: algdep(z, 4, use_bits=25)
x^2 - 2
sage: algdep(z, 4, use_digits=8)
x^2 - 2

Using the height_bound and proof parameters, we can see that \(pi\) is not the root of an integer polynomial of degree at most 5 and coefficients bounded above by 10:

sage: algdep(pi.n(), 5, height_bound=10, proof=True) is None
True

For stronger results, we need more precision:

sage: algdep(pi.n(), 5, height_bound=100, proof=True) is None
Traceback (most recent call last):
...
ValueError: insufficient precision for non-existence proof
sage: algdep(pi.n(200), 5, height_bound=100, proof=True) is None
True

sage: algdep(pi.n(), 10, height_bound=10, proof=True) is None
Traceback (most recent call last):
...
ValueError: insufficient precision for non-existence proof
sage: algdep(pi.n(200), 10, height_bound=10, proof=True) is None
True

We can also use proof=True to get positive results:

sage: a = sqrt(2) + sqrt(3) + sqrt(5)
sage: algdep(a.n(), 8, height_bound=1000, proof=True)
Traceback (most recent call last):
...
ValueError: insufficient precision for uniqueness proof
sage: f = algdep(a.n(1000), 8, height_bound=1000, proof=True); f
x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576
sage: f(a).expand()
0

sage.arith.misc.algebraic_dependency(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False)

Return an irreducible polynomial of degree at most \(degree\) which is approximately satisfied by the number \(z\).

You can specify the number of known bits or digits of \(z\) with known_bits=k or known_digits=k. PARI is then told to compute the result using \(0.8k\) of these bits/digits. Or, you can specify the precision to use directly with use_bits=k or use_digits=k. If none of these are specified, then the precision is taken from the input value.

A height bound may be specified to indicate the maximum coefficient size of the returned polynomial; if a sufficiently small polynomial is not found, then None will be returned. If proof=True then the result is returned only if it can be proved correct (i.e. the only possible minimal polynomial satisfying the height bound, or no such polynomial exists). Otherwise a ValueError is raised indicating that higher precision is required.

ALGORITHM: Uses LLL for real/complex inputs, PARI C-library algdep command otherwise.

Note that algebraic_dependency is a synonym for algdep.

INPUT:

• z - real, complex, or \(p\)-adic number
• degree - an integer

• height_bound - an integer (default: None) specifying the maximum

  coefficient size for the returned polynomial

• proof - a boolean (default: False), requires height_bound to be set

EXAMPLES:

sage: algdep(1.888888888888888, 1)
9*x - 17
sage: algdep(0.12121212121212,1)
33*x - 4
sage: algdep(sqrt(2),2)
x^2 - 2

This example involves a complex number:

sage: z = (1/2)*(1 + RDF(sqrt(3)) *CC.0); z
0.500000000000000 + 0.866025403784439*I
sage: algdep(z, 6)
x^2 - x + 1

This example involves a \(p\)-adic number:

sage: K = Qp(3, print_mode = 'series')
sage: a = K(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: algdep(a, 1)
19*x - 7

These examples show the importance of proper precision control. We compute a 200-bit approximation to \(sqrt(2)\) which is wrong in the 33’rd bit:

sage: z = sqrt(RealField(200)(2)) + (1/2)^33
sage: p = algdep(z, 4); p
227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088
sage: factor(p)
227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088
sage: algdep(z, 4, known_bits=32)
x^2 - 2
sage: algdep(z, 4, known_digits=10)
x^2 - 2
sage: algdep(z, 4, use_bits=25)
x^2 - 2
sage: algdep(z, 4, use_digits=8)
x^2 - 2

Using the height_bound and proof parameters, we can see that \(pi\) is not the root of an integer polynomial of degree at most 5 and coefficients bounded above by 10:

sage: algdep(pi.n(), 5, height_bound=10, proof=True) is None
True

For stronger results, we need more precision:

sage: algdep(pi.n(), 5, height_bound=100, proof=True) is None
Traceback (most recent call last):
...
ValueError: insufficient precision for non-existence proof
sage: algdep(pi.n(200), 5, height_bound=100, proof=True) is None
True

sage: algdep(pi.n(), 10, height_bound=10, proof=True) is None
Traceback (most recent call last):
...
ValueError: insufficient precision for non-existence proof
sage: algdep(pi.n(200), 10, height_bound=10, proof=True) is None
True

We can also use proof=True to get positive results:

sage: a = sqrt(2) + sqrt(3) + sqrt(5)
sage: algdep(a.n(), 8, height_bound=1000, proof=True)
Traceback (most recent call last):
...
ValueError: insufficient precision for uniqueness proof
sage: f = algdep(a.n(1000), 8, height_bound=1000, proof=True); f
x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576
sage: f(a).expand()
0

sage.arith.misc.bernoulli(n, algorithm='default', num_threads=1)

Return the n-th Bernoulli number, as a rational number.

INPUT:

• n - an integer
• algorithm:
  • 'default' – use ‘flint’ for n <= 20000, then ‘arb’ for n <= 300000 and ‘bernmm’ for larger values (this is just a heuristic, and not guaranteed to be optimal on all hardware)
  • 'arb' – use the arb library
  • 'flint' – use the FLINT library
  • 'pari' – use the PARI C library
  • 'gap' – use GAP
  • 'gp' – use PARI/GP interpreter
  • 'magma' – use MAGMA (optional)
  • 'bernmm' – use bernmm package (a multimodular algorithm)
• num_threads - positive integer, number of threads to use (only used for bernmm algorithm)

EXAMPLES:

sage: bernoulli(12)
-691/2730
sage: bernoulli(50)
495057205241079648212477525/66

We demonstrate each of the alternative algorithms:

sage: bernoulli(12, algorithm='arb')
-691/2730
sage: bernoulli(12, algorithm='flint')
-691/2730
sage: bernoulli(12, algorithm='gap')
-691/2730
sage: bernoulli(12, algorithm='gp')
-691/2730
sage: bernoulli(12, algorithm='magma')           # optional - magma
-691/2730
sage: bernoulli(12, algorithm='pari')
-691/2730
sage: bernoulli(12, algorithm='bernmm')
-691/2730
sage: bernoulli(12, algorithm='bernmm', num_threads=4)
-691/2730

AUTHOR:

• David Joyner and William Stein

sage.arith.misc.binomial(x, m, **kwds)

Return the binomial coefficient

\[\binom{x}{m} = x (x-1) \cdots (x-m+1) / m!\]

which is defined for \(m \in \ZZ\) and any \(x\). We extend this definition to include cases when \(x-m\) is an integer but \(m\) is not by

\[\binom{x}{m} = \binom{x}{x-m}\]

If \(m < 0\), return \(0\).

INPUT:

• x, m - numbers or symbolic expressions. Either m or x-m must be an integer.

OUTPUT: number or symbolic expression (if input is symbolic)

EXAMPLES:

sage: from sage.arith.misc import binomial
sage: binomial(5,2)
10
sage: binomial(2,0)
1
sage: binomial(1/2, 0)
1
sage: binomial(3,-1)
0
sage: binomial(20,10)
184756
sage: binomial(-2, 5)
-6
sage: binomial(-5, -2)
0
sage: binomial(RealField()('2.5'), 2)
1.87500000000000
sage: n=var('n'); binomial(n,2)
1/2*(n - 1)*n
sage: n=var('n'); binomial(n,n)
1
sage: n=var('n'); binomial(n,n-1)
n
sage: binomial(2^100, 2^100)
1

sage: x = polygen(ZZ)
sage: binomial(x, 3)
1/6*x^3 - 1/2*x^2 + 1/3*x
sage: binomial(x, x-3)
1/6*x^3 - 1/2*x^2 + 1/3*x

If \(x \in \ZZ\), there is an optional ‘algorithm’ parameter, which can be ‘mpir’ (faster for small values) or ‘pari’ (faster for large values):

sage: a = binomial(100, 45, algorithm='mpir')
sage: b = binomial(100, 45, algorithm='pari')
sage: a == b
True

sage.arith.misc.binomial_coefficients(n)

Return a dictionary containing pairs \(\{(k_1,k_2) : C_{k,n}\}\) where \(C_{k_n}\) are binomial coefficients and \(n = k_1 + k_2\).

INPUT:

• n - an integer

OUTPUT: dict

EXAMPLES:

sage: sorted(binomial_coefficients(3).items())
[((0, 3), 1), ((1, 2), 3), ((2, 1), 3), ((3, 0), 1)]

Notice the coefficients above are the same as below:

sage: R.<x,y> = QQ[]
sage: (x+y)^3
x^3 + 3*x^2*y + 3*x*y^2 + y^3

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: sorted(binomial_coefficients(int8(3)).items())
[((0, 3), 1), ((1, 2), 3), ((2, 1), 3), ((3, 0), 1)]
sage: from gmpy2 import mpz
sage: sorted(binomial_coefficients(mpz(3)).items())
[((0, 3), 1), ((1, 2), 3), ((2, 1), 3), ((3, 0), 1)]

AUTHORS:

• Fredrik Johansson

sage.arith.misc.continuant(v, n=None)

Function returns the continuant of the sequence \(v\) (list or tuple).

Definition: see Graham, Knuth and Patashnik, Concrete Mathematics, section 6.7: Continuants. The continuant is defined by

• \(K_0() = 1\)
• \(K_1(x_1) = x_1\)
• \(K_n(x_1, \cdots, x_n) = K_{n-1}(x_n, \cdots x_{n-1})x_n + K_{n-2}(x_1, \cdots, x_{n-2})\)

If n = None or n > len(v) the default n = len(v) is used.

INPUT:

• v - list or tuple of elements of a ring
• n - optional integer

OUTPUT: element of ring (integer, polynomial, etcetera).

EXAMPLES:

sage: continuant([1,2,3])
10
sage: p = continuant([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10])
sage: q = continuant([1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10])
sage: p/q
517656/190435
sage: continued_fraction([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10]).convergent(14)
517656/190435
sage: x = PolynomialRing(RationalField(),'x',5).gens()
sage: continuant(x)
x0*x1*x2*x3*x4 + x0*x1*x2 + x0*x1*x4 + x0*x3*x4 + x2*x3*x4 + x0 + x2 + x4
sage: continuant(x, 3)
x0*x1*x2 + x0 + x2
sage: continuant(x,2)
x0*x1 + 1

We verify the identity

\[K_n(z,z,\cdots,z) = \sum_{k=0}^n \binom{n-k}{k} z^{n-2k}\]

for \(n = 6\) using polynomial arithmetic:

sage: z = QQ['z'].0
sage: continuant((z,z,z,z,z,z,z,z,z,z,z,z,z,z,z),6)
z^6 + 5*z^4 + 6*z^2 + 1

sage: continuant(9)
Traceback (most recent call last):
...
TypeError: object of type 'sage.rings.integer.Integer' has no len()

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: continuant([int8(1),int8(2),int8(3)])
10
sage: from gmpy2 import mpz
sage: continuant([mpz(1),mpz(2),mpz(3)])
mpz(10)

AUTHORS:

• Jaap Spies (2007-02-06)

sage.arith.misc.crt(a, b, m=None, n=None)

Return a solution to a Chinese Remainder Theorem problem.

INPUT:

• a, b - two residues (elements of some ring for which extended gcd is available), or two lists, one of residues and one of moduli.
• m, n - (default: None) two moduli, or None.

OUTPUT:

If m, n are not None, returns a solution \(x\) to the simultaneous congruences \(x\equiv a \bmod m\) and \(x\equiv b \bmod n\), if one exists. By the Chinese Remainder Theorem, a solution to the simultaneous congruences exists if and only if \(a\equiv b\pmod{\gcd(m,n)}\). The solution \(x\) is only well-defined modulo \(\text{lcm}(m,n)\).

If a and b are lists, returns a simultaneous solution to the congruences \(x\equiv a_i\pmod{b_i}\), if one exists.

See also

• CRT_list()

EXAMPLES:

Using crt by giving it pairs of residues and moduli:

sage: crt(2, 1, 3, 5)
11
sage: crt(13, 20, 100, 301)
28013
sage: crt([2, 1], [3, 5])
11
sage: crt([13, 20], [100, 301])
28013

You can also use upper case:

sage: c = CRT(2,3, 3, 5); c
8
sage: c % 3 == 2
True
sage: c % 5 == 3
True

Note that this also works for polynomial rings:

sage: K.<a> = NumberField(x^3 - 7)
sage: R.<y> = K[]
sage: f = y^2 + 3
sage: g = y^3 - 5
sage: CRT(1,3,f,g)
-3/26*y^4 + 5/26*y^3 + 15/26*y + 53/26
sage: CRT(1,a,f,g)
(-3/52*a + 3/52)*y^4 + (5/52*a - 5/52)*y^3 + (15/52*a - 15/52)*y + 27/52*a + 25/52

You can also do this for any number of moduli:

sage: K.<a> = NumberField(x^3 - 7)
sage: R.<x> = K[]
sage: CRT([], [])
0
sage: CRT([a], [x])
a
sage: f = x^2 + 3
sage: g = x^3 - 5
sage: h = x^5 + x^2 - 9
sage: k = CRT([1, a, 3], [f, g, h]); k
(127/26988*a - 5807/386828)*x^9 + (45/8996*a - 33677/1160484)*x^8 + (2/173*a - 6/173)*x^7 + (133/6747*a - 5373/96707)*x^6 + (-6/2249*a + 18584/290121)*x^5 + (-277/8996*a + 38847/386828)*x^4 + (-135/4498*a + 42673/193414)*x^3 + (-1005/8996*a + 470245/1160484)*x^2 + (-1215/8996*a + 141165/386828)*x + 621/8996*a + 836445/386828
sage: k.mod(f)
1
sage: k.mod(g)
a
sage: k.mod(h)
3

If the moduli are not coprime, a solution may not exist:

sage: crt(4,8,8,12)
20
sage: crt(4,6,8,12)
Traceback (most recent call last):
...
ValueError: No solution to crt problem since gcd(8,12) does not divide 4-6

sage: x = polygen(QQ)
sage: crt(2,3,x-1,x+1)
-1/2*x + 5/2
sage: crt(2,x,x^2-1,x^2+1)
-1/2*x^3 + x^2 + 1/2*x + 1
sage: crt(2,x,x^2-1,x^3-1)
Traceback (most recent call last):
...
ValueError: No solution to crt problem since gcd(x^2 - 1,x^3 - 1) does not divide 2-x

sage: crt(int(2), int(3), int(7), int(11))
58

crt also work with numpy and gmpy2 numbers:

sage: import numpy
sage: crt(numpy.int8(2), numpy.int8(3), numpy.int8(7), numpy.int8(11))
58
sage: from gmpy2 import mpz
sage: crt(mpz(2), mpz(3), mpz(7), mpz(11))
58
sage: crt(mpz(2), 3, mpz(7), numpy.int8(11))
58

sage.arith.misc.dedekind_psi(N)

Return the value of the Dedekind psi function at N.

INPUT:

• N – a positive integer

OUTPUT:

an integer

The Dedekind psi function is the multiplicative function defined by

\[\psi(n) = n \prod_{p|n, p prime} (1 + 1/p).\]

See Wikipedia article Dedekind_psi_function and OEIS sequence A001615.

EXAMPLES:

sage: from sage.arith.misc import dedekind_psi
sage: [dedekind_psi(d) for d in range(1, 12)]
[1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12]

sage.arith.misc.dedekind_sum(p, q, algorithm='default')

Return the Dedekind sum \(s(p,q)\) defined for integers \(p\), \(q\) as

\[s(p,q) = \sum_{i=0}^{q-1} \left(\!\left(\frac{i}{q}\right)\!\right) \left(\!\left(\frac{pi}{q}\right)\!\right)\]

where

\[\begin{split}((x))=\begin{cases} x-\lfloor x \rfloor - \frac{1}{2} &\mbox{if } x \in \QQ \setminus \ZZ \\ 0 & \mbox{if } x \in \ZZ. \end{cases}\end{split}\]

Warning

Caution is required as the Dedekind sum sometimes depends on the algorithm or is left undefined when \(p\) and \(q\) are not coprime.

INPUT:

• p, q – integers
• algorithm – must be one of the following
  • 'default' - (default) use FLINT
  • 'flint' - use FLINT
  • 'pari' - use PARI (gives different results if \(p\) and \(q\) are not coprime)

OUTPUT: a rational number

EXAMPLES:

Several small values:

sage: for q in range(10): print([dedekind_sum(p,q) for p in range(q+1)])
[0]
[0, 0]
[0, 0, 0]
[0, 1/18, -1/18, 0]
[0, 1/8, 0, -1/8, 0]
[0, 1/5, 0, 0, -1/5, 0]
[0, 5/18, 1/18, 0, -1/18, -5/18, 0]
[0, 5/14, 1/14, -1/14, 1/14, -1/14, -5/14, 0]
[0, 7/16, 1/8, 1/16, 0, -1/16, -1/8, -7/16, 0]
[0, 14/27, 4/27, 1/18, -4/27, 4/27, -1/18, -4/27, -14/27, 0]

Check relations for restricted arguments:

sage: q = 23; dedekind_sum(1, q); (q-1)*(q-2)/(12*q)
77/46
77/46
sage: p, q = 100, 723    # must be coprime
sage: dedekind_sum(p, q) + dedekind_sum(q, p)
31583/86760
sage: -1/4 + (p/q + q/p + 1/(p*q))/12
31583/86760

We check that evaluation works with large input:

sage: dedekind_sum(3^54 - 1, 2^93 + 1)
459340694971839990630374299870/29710560942849126597578981379
sage: dedekind_sum(3^54 - 1, 2^93 + 1, algorithm='pari')
459340694971839990630374299870/29710560942849126597578981379

We check consistency of the results:

sage: dedekind_sum(5, 7, algorithm='default')
-1/14
sage: dedekind_sum(5, 7, algorithm='flint')
-1/14
sage: dedekind_sum(5, 7, algorithm='pari')
-1/14
sage: dedekind_sum(6, 8, algorithm='default')
-1/8
sage: dedekind_sum(6, 8, algorithm='flint')
-1/8
sage: dedekind_sum(6, 8, algorithm='pari')
-1/8

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: dedekind_sum(int8(5), int8(7), algorithm='default')
-1/14
sage: from gmpy2 import mpz
sage: dedekind_sum(mpz(5), mpz(7), algorithm='default')
-1/14

REFERENCES:

• [Ap1997]
• Wikipedia article Dedekind_sum

sage.arith.misc.differences(lis, n=1)

Return the \(n\) successive differences of the elements in \(lis\).

EXAMPLES:

sage: differences(prime_range(50))
[1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4]
sage: differences([i^2 for i in range(1,11)])
[3, 5, 7, 9, 11, 13, 15, 17, 19]
sage: differences([i^3 + 3*i for i in range(1,21)])
[10, 22, 40, 64, 94, 130, 172, 220, 274, 334, 400, 472, 550, 634, 724, 820, 922, 1030, 1144]
sage: differences([i^3 - i^2 for i in range(1,21)], 2)
[10, 16, 22, 28, 34, 40, 46, 52, 58, 64, 70, 76, 82, 88, 94, 100, 106, 112]
sage: differences([p - i^2 for i, p in enumerate(prime_range(50))], 3)
[-1, 2, -4, 4, -4, 4, 0, -6, 8, -6, 0, 4]

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: differences([int8(1),int8(4),int8(6),int8(19)])
[3, 2, 13]
sage: from gmpy2 import mpz
sage: differences([mpz(1),mpz(4),mpz(6),mpz(19)])
[mpz(3), mpz(2), mpz(13)]

AUTHORS:

• Timothy Clemans (2008-03-09)

sage.arith.misc.divisors(n)

Return the list of all divisors (up to units) of this element of a unique factorization domain.

For an integer, the list of all positive integer divisors of this integer, sorted in increasing order, is returned.

INPUT:

• n - the element

EXAMPLES:

Divisors of integers:

sage: divisors(-3)
[1, 3]
sage: divisors(6)
[1, 2, 3, 6]
sage: divisors(28)
[1, 2, 4, 7, 14, 28]
sage: divisors(2^5)
[1, 2, 4, 8, 16, 32]
sage: divisors(100)
[1, 2, 4, 5, 10, 20, 25, 50, 100]
sage: divisors(1)
[1]
sage: divisors(0)
Traceback (most recent call last):
...
ValueError: n must be nonzero
sage: divisors(2^3 * 3^2 * 17)
[1, 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72,
102, 136, 153, 204, 306, 408, 612, 1224]

This function works whenever one has unique factorization:

sage: K.<a> = QuadraticField(7)
sage: divisors(K.ideal(7))
[Fractional ideal (1), Fractional ideal (a), Fractional ideal (7)]
sage: divisors(K.ideal(3))
[Fractional ideal (1), Fractional ideal (3),
Fractional ideal (-a + 2), Fractional ideal (-a - 2)]
sage: divisors(K.ideal(35))
[Fractional ideal (1), Fractional ideal (5), Fractional ideal (a),
Fractional ideal (7), Fractional ideal (5*a), Fractional ideal (35)]

sage.arith.misc.eratosthenes(n)

Return a list of the primes \(\leq n\).

This is extremely slow and is for educational purposes only.

INPUT:

• n - a positive integer

OUTPUT:

• a list of primes less than or equal to n.

EXAMPLES:

sage: eratosthenes(3)
[2, 3]
sage: eratosthenes(50)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
sage: len(eratosthenes(100))
25
sage: eratosthenes(213) == prime_range(213)
True

sage.arith.misc.factor(n, proof=None, int_=False, algorithm='pari', verbose=0, **kwds)

Return the factorization of n. The result depends on the type of n.

If n is an integer, returns the factorization as an object of type Factorization.

If n is not an integer, n.factor(proof=proof, **kwds) gets called. See n.factor?? for more documentation in this case.

Warning

This means that applying factor to an integer result of a symbolic computation will not factor the integer, because it is considered as an element of a larger symbolic ring.

EXAMPLES:

sage: f(n)=n^2
sage: is_prime(f(3))
False
sage: factor(f(3))
9

INPUT:

• n - an nonzero integer
• proof - bool or None (default: None)
• int_ - bool (default: False) whether to return answers as Python ints
• algorithm - string
  • 'pari' - (default) use the PARI c library
  • 'kash' - use KASH computer algebra system (requires the optional kash package be installed)
  • 'magma' - use Magma (requires magma be installed)
• verbose - integer (default: 0); PARI’s debug variable is set to this; e.g., set to 4 or 8 to see lots of output during factorization.

OUTPUT:

• factorization of n

The qsieve and ecm commands give access to highly optimized implementations of algorithms for doing certain integer factorization problems. These implementations are not used by the generic factor command, which currently just calls PARI (note that PARI also implements sieve and ecm algorithms, but they are not as optimized). Thus you might consider using them instead for certain numbers.

The factorization returned is an element of the class Factorization; see Factorization?? for more details, and examples below for usage. A Factorization contains both the unit factor (+1 or -1) and a sorted list of (prime, exponent) pairs.

The factorization displays in pretty-print format but it is easy to obtain access to the (prime,exponent) pairs and the unit, to recover the number from its factorization, and even to multiply two factorizations. See examples below.

EXAMPLES:

sage: factor(500)
2^2 * 5^3
sage: factor(-20)
-1 * 2^2 * 5
sage: f=factor(-20)
sage: list(f)
[(2, 2), (5, 1)]
sage: f.unit()
-1
sage: f.value()
-20
sage: factor( -next_prime(10^2) * next_prime(10^7) )
-1 * 101 * 10000019

sage: factor(-500, algorithm='kash')      # optional - kash
-1 * 2^2 * 5^3

sage: factor(-500, algorithm='magma')     # optional - magma
-1 * 2^2 * 5^3

sage: factor(0)
Traceback (most recent call last):
...
ArithmeticError: factorization of 0 is not defined
sage: factor(1)
1
sage: factor(-1)
-1
sage: factor(2^(2^7)+1)
59649589127497217 * 5704689200685129054721

Sage calls PARI’s factor, which has proof False by default. Sage has a global proof flag, set to True by default (see sage.structure.proof.proof, or proof.[tab]). To override the default, call this function with proof=False.

sage: factor(3^89-1, proof=False)
2 * 179 * 1611479891519807 * 5042939439565996049162197

sage: factor(2^197 + 1)  # long time (2s)
3 * 197002597249 * 1348959352853811313 * 251951573867253012259144010843

Any object which has a factor method can be factored like this:

sage: K.<i> = QuadraticField(-1)
sage: factor(122 - 454*i)
(-3*i - 2) * (-i - 2)^3 * (i + 1)^3 * (i + 4)

To access the data in a factorization:

sage: f = factor(420); f
2^2 * 3 * 5 * 7
sage: [x for x in f]
[(2, 2), (3, 1), (5, 1), (7, 1)]
sage: [p for p,e in f]
[2, 3, 5, 7]
sage: [e for p,e in f]
[2, 1, 1, 1]
sage: [p^e for p,e in f]
[4, 3, 5, 7]

We can factor Python, numpy and gmpy2 numbers:

sage: factor(math.pi)
3.141592653589793
sage: import numpy
sage: factor(numpy.int8(30))
2 * 3 * 5
sage: import gmpy2
sage: factor(gmpy2.mpz(30))
2 * 3 * 5

sage.arith.misc.factorial(n, algorithm='gmp')

Compute the factorial of \(n\), which is the product \(1\cdot 2\cdot 3 \cdots (n-1)\cdot n\).

INPUT:

• n - an integer
• algorithm - string (default: ‘gmp’):
  • 'gmp' - use the GMP C-library factorial function
  • 'pari' - use PARI’s factorial function

OUTPUT: an integer

EXAMPLES:

sage: from sage.arith.misc import factorial
sage: factorial(0)
1
sage: factorial(4)
24
sage: factorial(10)
3628800
sage: factorial(1) == factorial(0)
True
sage: factorial(6) == 6*5*4*3*2
True
sage: factorial(1) == factorial(0)
True
sage: factorial(71) == 71* factorial(70)
True
sage: factorial(-32)
Traceback (most recent call last):
...
ValueError: factorial -- must be nonnegative

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: factorial(int8(4))
24
sage: from gmpy2 import mpz
sage: factorial(mpz(4))
24

PERFORMANCE: This discussion is valid as of April 2006. All timings below are on a Pentium Core Duo 2Ghz MacBook Pro running Linux with a 2.6.16.1 kernel.

• It takes less than a minute to compute the factorial of \(10^7\) using the GMP algorithm, and the factorial of \(10^6\) takes less than 4 seconds.
• The GMP algorithm is faster and more memory efficient than the PARI algorithm. E.g., PARI computes \(10^7\) factorial in 100 seconds on the core duo 2Ghz.
• For comparison, computation in Magma \(\leq\) 2.12-10 of \(n!\) is best done using *[1..n]. It takes 113 seconds to compute the factorial of \(10^7\) and 6 seconds to compute the factorial of \(10^6\). Mathematica V5.2 compute the factorial of \(10^7\) in 136 seconds and the factorial of \(10^6\) in 7 seconds. (Mathematica is notably very efficient at memory usage when doing factorial calculations.)

sage.arith.misc.falling_factorial(x, a)

Return the falling factorial \((x)_a\).

The notation in the literature is a mess: often \((x)_a\), but there are many other notations: GKP: Concrete Mathematics uses \(x^{\underline{a}}\).

Definition: for integer \(a \ge 0\) we have \(x(x-1) \cdots (x-a+1)\). In all other cases we use the GAMMA-function: \(\frac {\Gamma(x+1)} {\Gamma(x-a+1)}\).

INPUT:

• x – element of a ring
• a – a non-negative integer or
• x and a – any numbers

OUTPUT: the falling factorial

See also

rising_factorial()

EXAMPLES:

sage: falling_factorial(10, 3)
720
sage: falling_factorial(10, RR('3.0'))
720.000000000000
sage: falling_factorial(10, RR('3.3'))
1310.11633396601
sage: falling_factorial(10, 10)
3628800
sage: factorial(10)
3628800
sage: a = falling_factorial(1+I, I); a
gamma(I + 2)
sage: CC(a)
0.652965496420167 + 0.343065839816545*I
sage: falling_factorial(1+I, 4)
4*I + 2
sage: falling_factorial(I, 4)
-10

sage: M = MatrixSpace(ZZ, 4, 4)
sage: A = M([1,0,1,0,1,0,1,0,1,0,10,10,1,0,1,1])
sage: falling_factorial(A, 2) # A(A - I)
[  1   0  10  10]
[  1   0  10  10]
[ 20   0 101 100]
[  2   0  11  10]

sage: x = ZZ['x'].0
sage: falling_factorial(x, 4)
x^4 - 6*x^3 + 11*x^2 - 6*x

AUTHORS:

• Jaap Spies (2006-03-05)

sage.arith.misc.four_squares(n)

Write the integer \(n\) as a sum of four integer squares.

INPUT:

• n – an integer

OUTPUT: a tuple \((a,b,c,d)\) of non-negative integers such that \(n = a^2 + b^2 + c^2 + d^2\) with \(a <= b <= c <= d\).

EXAMPLES:

sage: four_squares(3)
(0, 1, 1, 1)
sage: four_squares(13)
(0, 0, 2, 3)
sage: four_squares(130)
(0, 0, 3, 11)
sage: four_squares(1101011011004)
(90, 102, 1220, 1049290)
sage: four_squares(10^100-1)
(155024616290, 2612183768627, 14142135623730950488016887, 99999999999999999999999999999999999999999999999999)
sage: for i in range(2^129, 2^129+10000):  # long time
....:     S = four_squares(i)
....:     assert sum(x^2 for x in S) == i

sage.arith.misc.fundamental_discriminant(D)

Return the discriminant of the quadratic extension \(K=Q(\sqrt{D})\), i.e. an integer d congruent to either 0 or 1, mod 4, and such that, at most, the only square dividing it is 4.

INPUT:

• D - an integer

OUTPUT:

• an integer, the fundamental discriminant

EXAMPLES:

sage: fundamental_discriminant(102)
408
sage: fundamental_discriminant(720)
5
sage: fundamental_discriminant(2)
8

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: fundamental_discriminant(int8(102))
408
sage: from gmpy2 import mpz
sage: fundamental_discriminant(mpz(102))
408

sage.arith.misc.gauss_sum(char_value, finite_field)

Return the Gauss sums for a general finite field.

INPUT:

• char_value – choice of multiplicative character, given by its value on the finite_field.multiplicative_generator()
• finite_field – a finite field

OUTPUT:

an element of the parent ring of char_value, that can be any field containing enough roots of unity, for example the UniversalCyclotomicField, QQbar or ComplexField

For a finite field \(F\) of characteristic \(p\), the Gauss sum associated to a multiplicative character \(\chi\) (with values in a ring \(K\)) is defined as

\[\sum_{x \in F^{\times}} \chi(x) \zeta_p^{\operatorname{Tr} x},\]

where \(\zeta_p \in K\) is a primitive root of unity of order \(p\) and Tr is the trace map from \(F\) to its prime field \(\GF{p}\).

For more info on Gauss sums, see Wikipedia article Gauss_sum.

Todo

Implement general Gauss sums for an arbitrary pair (multiplicative_character, additive_character)

EXAMPLES:

sage: from sage.arith.misc import gauss_sum
sage: F = GF(5); q = 5
sage: zq = UniversalCyclotomicField().zeta(q-1)
sage: L = [gauss_sum(zq**i,F) for i in range(5)]; L
[-1,
 E(20)^4 + E(20)^13 - E(20)^16 - E(20)^17,
 E(5) - E(5)^2 - E(5)^3 + E(5)^4,
 E(20)^4 - E(20)^13 - E(20)^16 + E(20)^17,
 -1]
sage: [g*g.conjugate() for g in L]
[1, 5, 5, 5, 1]

sage: F = GF(11**2); q = 11**2
sage: zq = UniversalCyclotomicField().zeta(q-1)
sage: g = gauss_sum(zq**4,F)
sage: g*g.conjugate()
121

See also

• sage.rings.padics.misc.gauss_sum() for a \(p\)-adic version
• sage.modular.dirichlet.DirichletCharacter.gauss_sum() for prime finite fields
• sage.modular.dirichlet.DirichletCharacter.gauss_sum_numerical() for prime finite fields

sage.arith.misc.gcd(a, b=None, **kwargs)

Return the greatest common divisor of a and b.

If a is a list and b is omitted, return instead the greatest common divisor of all elements of a.

INPUT:

• a,b – two elements of a ring with gcd or
• a – a list or tuple of elements of a ring with gcd

Additional keyword arguments are passed to the respectively called methods.

OUTPUT:

The given elements are first coerced into a common parent. Then, their greatest common divisor in that common parent is returned.

EXAMPLES:

sage: GCD(97,100)
1
sage: GCD(97*10^15, 19^20*97^2)
97
sage: GCD(2/3, 4/5)
2/15
sage: GCD([2,4,6,8])
2
sage: GCD(srange(0,10000,10))  # fast  !!
10

Note that to take the gcd of \(n\) elements for \(n \not= 2\) you must put the elements into a list by enclosing them in [..]. Before trac ticket #4988 the following wrongly returned 3 since the third parameter was just ignored:

sage: gcd(3, 6, 2)
Traceback (most recent call last):
...
TypeError: gcd() takes ...
sage: gcd([3, 6, 2])
1

Similarly, giving just one element (which is not a list) gives an error:

sage: gcd(3)
Traceback (most recent call last):
...
TypeError: 'sage.rings.integer.Integer' object is not iterable

By convention, the gcd of the empty list is (the integer) 0:

sage: gcd([])
0
sage: type(gcd([]))
<type 'sage.rings.integer.Integer'>

sage.arith.misc.get_gcd(order)

Return the fastest gcd function for integers of size no larger than order.

EXAMPLES:

sage: sage.arith.misc.get_gcd(4000)
<built-in method gcd_int of sage.rings.fast_arith.arith_int object at ...>
sage: sage.arith.misc.get_gcd(400000)
<built-in method gcd_longlong of sage.rings.fast_arith.arith_llong object at ...>
sage: sage.arith.misc.get_gcd(4000000000)
<function gcd at ...>

sage.arith.misc.get_inverse_mod(order)

Return the fastest inverse_mod function for integers of size no larger than order.

EXAMPLES:

sage: sage.arith.misc.get_inverse_mod(6000)
<built-in method inverse_mod_int of sage.rings.fast_arith.arith_int object at ...>
sage: sage.arith.misc.get_inverse_mod(600000)
<built-in method inverse_mod_longlong of sage.rings.fast_arith.arith_llong object at ...>
sage: sage.arith.misc.get_inverse_mod(6000000000)
<function inverse_mod at ...>

sage.arith.misc.hilbert_conductor(a, b)

This is the product of all (finite) primes where the Hilbert symbol is -1.

What is the same, this is the (reduced) discriminant of the quaternion algebra \((a,b)\) over \(\QQ\).

INPUT:

• a, b – integers

OUTPUT:

• squarefree positive integer

EXAMPLES:

sage: hilbert_conductor(-1, -1)
2
sage: hilbert_conductor(-1, -11)
11
sage: hilbert_conductor(-2, -5)
5
sage: hilbert_conductor(-3, -17)
17

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: hilbert_conductor(int8(-3), int8(-17))
17
sage: from gmpy2 import mpz
sage: hilbert_conductor(mpz(-3), mpz(-17))
17

AUTHOR:

• Gonzalo Tornaria (2009-03-02)

sage.arith.misc.hilbert_conductor_inverse(d)

Finds a pair of integers \((a,b)\) such that hilbert_conductor(a,b) == d.

The quaternion algebra \((a,b)\) over \(\QQ\) will then have (reduced) discriminant \(d\).

INPUT:

• d – square-free positive integer

OUTPUT: pair of integers

EXAMPLES:

sage: hilbert_conductor_inverse(2)
(-1, -1)
sage: hilbert_conductor_inverse(3)
(-1, -3)
sage: hilbert_conductor_inverse(6)
(-1, 3)
sage: hilbert_conductor_inverse(30)
(-3, -10)
sage: hilbert_conductor_inverse(4)
Traceback (most recent call last):
...
ValueError: d needs to be squarefree
sage: hilbert_conductor_inverse(-1)
Traceback (most recent call last):
...
ValueError: d needs to be positive

AUTHOR:

• Gonzalo Tornaria (2009-03-02)

sage.arith.misc.hilbert_symbol(a, b, p, algorithm='pari')

Return 1 if \(ax^2 + by^2\) \(p\)-adically represents a nonzero square, otherwise returns \(-1\). If either a or b is 0, returns 0.

INPUT:

• a, b - integers
• p - integer; either prime or -1 (which represents the archimedean place)
• algorithm - string
  • 'pari' - (default) use the PARI C library
  • 'direct' - use a Python implementation
  • 'all' - use both PARI and direct and check that the results agree, then return the common answer

OUTPUT: integer (0, -1, or 1)

EXAMPLES:

sage: hilbert_symbol (-1, -1, -1, algorithm='all')
-1
sage: hilbert_symbol (2,3, 5, algorithm='all')
1
sage: hilbert_symbol (4, 3, 5, algorithm='all')
1
sage: hilbert_symbol (0, 3, 5, algorithm='all')
0
sage: hilbert_symbol (-1, -1, 2, algorithm='all')
-1
sage: hilbert_symbol (1, -1, 2, algorithm='all')
1
sage: hilbert_symbol (3, -1, 2, algorithm='all')
-1

sage: hilbert_symbol(QQ(-1)/QQ(4), -1, 2) == -1
True
sage: hilbert_symbol(QQ(-1)/QQ(4), -1, 3) == 1
True

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: hilbert_symbol(int8(2),int8(3),int8(5),algorithm='all')
1
sage: from gmpy2 import mpz
sage: hilbert_symbol(mpz(2),mpz(3),mpz(5),algorithm='all')
1

AUTHORS:

• William Stein and David Kohel (2006-01-05)

sage.arith.misc.integer_ceil(x)

Return the ceiling of x.

EXAMPLES:

sage: integer_ceil(5.4)
6
sage: integer_ceil(x)
Traceback (most recent call last):
...
NotImplementedError: computation of ceil of x not implemented

Tests with numpy and gmpy2 numbers:

sage: from numpy import float32
sage: integer_ceil(float32(5.4))
6
sage: from gmpy2 import mpfr
sage: integer_ceil(mpfr(5.4))
6

sage.arith.misc.integer_floor(x)

Return the largest integer \(\leq x\).

INPUT:

• x - an object that has a floor method or is coercible to int

OUTPUT: an Integer

EXAMPLES:

sage: integer_floor(5.4)
5
sage: integer_floor(float(5.4))
5
sage: integer_floor(-5/2)
-3
sage: integer_floor(RDF(-5/2))
-3

sage: integer_floor(x)
Traceback (most recent call last):
...
NotImplementedError: computation of floor of x not implemented

Tests with numpy and gmpy2 numbers:

sage: from numpy import float32
sage: integer_floor(float32(5.4))
5
sage: from gmpy2 import mpfr
sage: integer_floor(mpfr(5.4))
5

sage.arith.misc.inverse_mod(a, m)

The inverse of the ring element a modulo m.

If no special inverse_mod is defined for the elements, it tries to coerce them into integers and perform the inversion there

sage: inverse_mod(7,1)
0
sage: inverse_mod(5,14)
3
sage: inverse_mod(3,-5)
2

Tests with numpy and mpz numbers:

sage: from numpy import int8
sage: inverse_mod(int8(5),int8(14))
3
sage: from gmpy2 import mpz
sage: inverse_mod(mpz(5),mpz(14))
3

sage.arith.misc.is_power_of_two(n)

Return whether n is a power of 2.

INPUT:

• n – integer

OUTPUT:

boolean

EXAMPLES:

sage: is_power_of_two(1024)
True
sage: is_power_of_two(1)
True
sage: is_power_of_two(24)
False
sage: is_power_of_two(0)
False
sage: is_power_of_two(-4)
False

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: is_power_of_two(int8(16))
True
sage: is_power_of_two(int8(24))
False
sage: from gmpy2 import mpz
sage: is_power_of_two(mpz(16))
True
sage: is_power_of_two(mpz(24))
False

sage.arith.misc.is_prime(n)

Return True if \(n\) is a prime number, and False otherwise.

Use a provable primality test or a strong pseudo-primality test depending on the global arithmetic proof flag.

INPUT:

• n - the object for which to determine primality

See also

• is_pseudoprime()
• sage.rings.integer.Integer.is_prime()

AUTHORS:

• Kevin Stueve kstueve@uw.edu (2010-01-17): delegated calculation to n.is_prime()

EXAMPLES:

sage: is_prime(389)
True
sage: is_prime(2000)
False
sage: is_prime(2)
True
sage: is_prime(-1)
False
sage: is_prime(1)
False
sage: is_prime(-2)
False

sage: a = 2**2048 + 981
sage: is_prime(a)    # not tested - takes ~ 1min
sage: proof.arithmetic(False)
sage: is_prime(a)    # instantaneous!
True
sage: proof.arithmetic(True)

sage.arith.misc.is_prime_power(n, get_data=False)

Test whether n is a positive power of a prime number

This function simply calls the method Integer.is_prime_power() of Integers.

INPUT:

• n – an integer
• get_data – if set to True, return a pair (p,k) such that this integer equals p^k instead of True or (self,0) instead of False

EXAMPLES:

sage: is_prime_power(389)
True
sage: is_prime_power(2000)
False
sage: is_prime_power(2)
True
sage: is_prime_power(1024)
True
sage: is_prime_power(1024, get_data=True)
(2, 10)

The same results can be obtained with:

sage: 389.is_prime_power()
True
sage: 2000.is_prime_power()
False
sage: 2.is_prime_power()
True
sage: 1024.is_prime_power()
True
sage: 1024.is_prime_power(get_data=True)
(2, 10)

sage.arith.misc.is_pseudoprime(n)

Test whether n is a pseudo-prime

The result is NOT proven correct - this is a pseudo-primality test!.

INPUT:

• n – an integer

Note

We do not consider negatives of prime numbers as prime.

EXAMPLES:

sage: is_pseudoprime(389)
True
sage: is_pseudoprime(2000)
False
sage: is_pseudoprime(2)
True
sage: is_pseudoprime(-1)
False
sage: factor(-6)
-1 * 2 * 3
sage: is_pseudoprime(1)
False
sage: is_pseudoprime(-2)
False

sage.arith.misc.is_pseudoprime_power(n, get_data=False)

Test if n is a power of a pseudoprime.

The result is NOT proven correct - this IS a pseudo-primality test!. Note that a prime power is a positive power of a prime number so that 1 is not a prime power.

INPUT:

• n - an integer
• get_data - (boolean) instead of a boolean return a pair \((p,k)\) so that n equals \(p^k\) and \(p\) is a pseudoprime or \((n,0)\) otherwise.

EXAMPLES:

sage: is_pseudoprime_power(389)
True
sage: is_pseudoprime_power(2000)
False
sage: is_pseudoprime_power(2)
True
sage: is_pseudoprime_power(1024)
True
sage: is_pseudoprime_power(-1)
False
sage: is_pseudoprime_power(1)
False
sage: is_pseudoprime_power(997^100)
True

Use of the get_data keyword:

sage: is_pseudoprime_power(3^1024, get_data=True)
(3, 1024)
sage: is_pseudoprime_power(2^256, get_data=True)
(2, 256)
sage: is_pseudoprime_power(31, get_data=True)
(31, 1)
sage: is_pseudoprime_power(15, get_data=True)
(15, 0)

Tests with numpy and gmpy2 numbers:

sage: from numpy import int16
sage: is_pseudoprime_power(int16(1024))
True
sage: from gmpy2 import mpz
sage: is_pseudoprime_power(mpz(1024))
True

sage.arith.misc.is_square(n, root=False)

Return whether or not n is square.

If n is a square also return the square root. If n is not square, also return None.

INPUT:

• n – an integer
• root – whether or not to also return a square root (default: False)

OUTPUT:

• bool – whether or not a square
• object – (optional) an actual square if found, and None otherwise.

EXAMPLES:

sage: is_square(2)
False
sage: is_square(4)
True
sage: is_square(2.2)
True
sage: is_square(-2.2)
False
sage: is_square(CDF(-2.2))
True
sage: is_square((x-1)^2)
Traceback (most recent call last):
...
NotImplementedError: is_square() not implemented for non numeric elements of Symbolic Ring

sage: is_square(4, True)
(True, 2)

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: is_square(int8(4))
True
sage: from gmpy2 import mpz
sage: is_square(mpz(4))
True

Tests with Polynomial:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: R.<a,b,c,d> = QQ[]
sage: p = a*b + c*d*a*d*a + 5
sage: is_square(p**2)
True

sage.arith.misc.is_squarefree(n)

Test whether n is square free.

EXAMPLES:

sage: is_squarefree(100)
False
sage: is_squarefree(101)
True

sage: R = ZZ['x']
sage: x = R.gen()
sage: is_squarefree((x^2+x+1) * (x-2))
True
sage: is_squarefree((x-1)**2 * (x-3))
False

sage: O = ZZ[sqrt(-1)]
sage: I = O.gen(1)
sage: is_squarefree(I+1)
True
sage: is_squarefree(O(2))
False
sage: O(2).factor()
(-I) * (I + 1)^2

This method fails on domains which are not Unique Factorization Domains:

sage: O = ZZ[sqrt(-5)]
sage: a = O.gen(1)
sage: is_squarefree(a - 3)
Traceback (most recent call last):
...
ArithmeticError: non-principal ideal in factorization

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: is_squarefree(int8(100))
False
sage: is_squarefree(int8(101))
True
sage: from gmpy2 import mpz
sage: is_squarefree(mpz(100))
False
sage: is_squarefree(mpz(101))
True

sage.arith.misc.jacobi_symbol(a, b)

The Jacobi symbol of integers a and b, where b is odd.

Note

The kronecker_symbol() command extends the Jacobi symbol to all integers b.

If

\(b = p_1^{e_1} * ... * p_r^{e_r}\)

then

\((a|b) = (a|p_1)^{e_1} ... (a|p_r)^{e_r}\)

where \((a|p_j)\) are Legendre Symbols.

INPUT:

• a - an integer
• b - an odd integer

EXAMPLES:

sage: jacobi_symbol(10,777)
-1
sage: jacobi_symbol(10,5)
0
sage: jacobi_symbol(10,2)
Traceback (most recent call last):
...
ValueError: second input must be odd, 2 is not odd

Tests with numpy and gmpy2 numbers:

sage: from numpy import int16
sage: jacobi_symbol(int16(10),int16(777))
-1
sage: from gmpy2 import mpz
sage: jacobi_symbol(mpz(10),mpz(777))
-1

sage.arith.misc.kronecker(x, y)

The Kronecker symbol \((x|y)\).

INPUT:

• x – integer
• y – integer

OUTPUT:

• an integer

EXAMPLES:

sage: kronecker_symbol(13,21)
-1
sage: kronecker_symbol(101,4)
1

This is also available as kronecker():

sage: kronecker(3,5)
-1
sage: kronecker(3,15)
0
sage: kronecker(2,15)
1
sage: kronecker(-2,15)
-1
sage: kronecker(2/3,5)
1

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: kronecker_symbol(int8(13),int8(21))
-1
sage: from gmpy2 import mpz
sage: kronecker_symbol(mpz(13),mpz(21))
-1

sage.arith.misc.kronecker_symbol(x, y)

The Kronecker symbol \((x|y)\).

INPUT:

• x – integer
• y – integer

OUTPUT:

• an integer

EXAMPLES:

sage: kronecker_symbol(13,21)
-1
sage: kronecker_symbol(101,4)
1

This is also available as kronecker():

sage: kronecker(3,5)
-1
sage: kronecker(3,15)
0
sage: kronecker(2,15)
1
sage: kronecker(-2,15)
-1
sage: kronecker(2/3,5)
1

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: kronecker_symbol(int8(13),int8(21))
-1
sage: from gmpy2 import mpz
sage: kronecker_symbol(mpz(13),mpz(21))
-1

sage.arith.misc.legendre_symbol(x, p)

The Legendre symbol \((x|p)\), for \(p\) prime.

Note

The kronecker_symbol() command extends the Legendre symbol to composite moduli and \(p=2\).

INPUT:

• x - integer
• p - an odd prime number

EXAMPLES:

sage: legendre_symbol(2,3)
-1
sage: legendre_symbol(1,3)
1
sage: legendre_symbol(1,2)
Traceback (most recent call last):
...
ValueError: p must be odd
sage: legendre_symbol(2,15)
Traceback (most recent call last):
...
ValueError: p must be a prime
sage: kronecker_symbol(2,15)
1
sage: legendre_symbol(2/3,7)
-1

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: legendre_symbol(int8(2),int8(3))
-1
sage: from gmpy2 import mpz
sage: legendre_symbol(mpz(2),mpz(3))
-1

sage.arith.misc.mqrr_rational_reconstruction(u, m, T)

Maximal Quotient Rational Reconstruction.

For research purposes only - this is pure Python, so slow.

INPUT:

• u, m, T - integers such that \(m > u \ge 0\), \(T > 0\).

OUTPUT:

Either integers \(n,d\) such that \(d>0\), \(\mathop{\mathrm{gcd}}(n,d)=1\), \(n/d=u \bmod m\), and \(T \cdot d \cdot |n| < m\), or None.

Reference: Monagan, Maximal Quotient Rational Reconstruction: An Almost Optimal Algorithm for Rational Reconstruction (page 11)

This algorithm is probabilistic.

EXAMPLES:

sage: mqrr_rational_reconstruction(21,3100,13)
(21, 1)

Tests with numpy and gmpy2 numbers:

sage: from numpy import int16
sage: mqrr_rational_reconstruction(int16(21),int16(3100),int16(13))
(21, 1)
sage: from gmpy2 import mpz
sage: mqrr_rational_reconstruction(mpz(21),mpz(3100),mpz(13))
(21, 1)

sage.arith.misc.multinomial(*ks)

Return the multinomial coefficient

INPUT:

• An arbitrary number of integer arguments \(k_1,\dots,k_n\)
• An iterable (e.g. a list) of integers \([k_1,\dots,k_n]\)

OUTPUT:

Return the integer:

\[\binom{k_1 + \cdots + k_n}{k_1, \cdots, k_n} =\frac{\left(\sum_{i=1}^n k_i\right)!}{\prod_{i=1}^n k_i!} = \prod_{i=1}^n \binom{\sum_{j=1}^i k_j}{k_i}\]

EXAMPLES:

sage: multinomial(0, 0, 2, 1, 0, 0)
3
sage: multinomial([0, 0, 2, 1, 0, 0])
3
sage: multinomial(3, 2)
10
sage: multinomial(2^30, 2, 1)
618970023101454657175683075
sage: multinomial([2^30, 2, 1])
618970023101454657175683075
sage: multinomial(Composition([1, 3]))
4
sage: multinomial(Partition([4, 2]))
15

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: multinomial(int8(3), int8(2))
10
sage: from gmpy2 import mpz
sage: multinomial(mpz(3), mpz(2))
mpz(10)

AUTHORS:

• Gabriel Ebner

sage.arith.misc.multinomial_coefficients(m, n)

Return a dictionary containing pairs \(\{(k_1, k_2, ..., k_m) : C_{k, n}\}\) where \(C_{k, n}\) are multinomial coefficients such that \(n = k_1 + k_2 + ...+ k_m\).

INPUT:

• m - integer
• n - integer

OUTPUT: dict

EXAMPLES:

sage: sorted(multinomial_coefficients(2, 5).items())
[((0, 5), 1), ((1, 4), 5), ((2, 3), 10), ((3, 2), 10), ((4, 1), 5), ((5, 0), 1)]

Notice that these are the coefficients of \((x+y)^5\):

sage: R.<x,y> = QQ[]
sage: (x+y)^5
x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + y^5

sage: sorted(multinomial_coefficients(3, 2).items())
[((0, 0, 2), 1), ((0, 1, 1), 2), ((0, 2, 0), 1), ((1, 0, 1), 2), ((1, 1, 0), 2), ((2, 0, 0), 1)]

ALGORITHM: The algorithm we implement for computing the multinomial coefficients is based on the following result:

\[\binom{n}{k_1, \cdots, k_m} = \frac{k_1+1}{n-k_1}\sum_{i=2}^m \binom{n}{k_1+1, \cdots, k_i-1, \cdots}\]

e.g.:

sage: k = (2, 4, 1, 0, 2, 6, 0, 0, 3, 5, 7, 1) # random value
sage: n = sum(k)
sage: s = 0
sage: for i in range(1, len(k)):
....:     ki = list(k)
....:     ki[0] += 1
....:     ki[i] -= 1
....:     s += multinomial(n, *ki)
sage: multinomial(n, *k) == (k[0] + 1) / (n - k[0]) * s
True

sage.arith.misc.next_prime(n, proof=None)

The next prime greater than the integer n. If n is prime, then this function does not return n, but the next prime after n. If the optional argument proof is False, this function only returns a pseudo-prime, as defined by the PARI nextprime function. If it is None, uses the global default (see sage.structure.proof.proof)

INPUT:

• n - integer
• proof - bool or None (default: None)

EXAMPLES:

sage: next_prime(-100)
2
sage: next_prime(1)
2
sage: next_prime(2)
3
sage: next_prime(3)
5
sage: next_prime(4)
5

Notice that the next_prime(5) is not 5 but 7.

sage: next_prime(5)
7
sage: next_prime(2004)
2011

sage.arith.misc.next_prime_power(n)

Return the smallest prime power greater than n.

Note that if n is a prime power, then this function does not return n, but the next prime power after n.

This function just calls the method Integer.next_prime_power() of Integers.

See also

• is_prime_power() (and Integer.is_prime_power())
• previous_prime_power() (and Integer.previous_prime_power())

EXAMPLES:

sage: next_prime_power(1)
2
sage: next_prime_power(2)
3
sage: next_prime_power(10)
11
sage: next_prime_power(7)
8
sage: next_prime_power(99)
101

The same results can be obtained with:

sage: 1.next_prime_power()
2
sage: 2.next_prime_power()
3
sage: 10.next_prime_power()
11

Note that \(2\) is the smallest prime power:

sage: next_prime_power(-10)
2
sage: next_prime_power(0)
2

sage.arith.misc.next_probable_prime(n)

Return the next probable prime after self, as determined by PARI.

INPUT:

• n - an integer

EXAMPLES:

sage: next_probable_prime(-100)
2
sage: next_probable_prime(19)
23
sage: next_probable_prime(int(999999999))
1000000007
sage: next_probable_prime(2^768)
1552518092300708935148979488462502555256886017116696611139052038026050952686376886330878408828646477950487730697131073206171580044114814391444287275041181139204454976020849905550265285631598444825262999193716468750892846853816058039

sage.arith.misc.nth_prime(n)

Return the n-th prime number (1-indexed, so that 2 is the 1st prime.)

INPUT:

• n – a positive integer

OUTPUT:

• the n-th prime number

EXAMPLES:

sage: nth_prime(3)
5
sage: nth_prime(10)
29
sage: nth_prime(10^7)
179424673

sage: nth_prime(0)
Traceback (most recent call last):
...
ValueError: nth prime meaningless for non-positive n (=0)

sage.arith.misc.number_of_divisors(n)

Return the number of divisors of the integer n.

INPUT:

• n - a nonzero integer

OUTPUT:

• an integer, the number of divisors of n

EXAMPLES:

sage: number_of_divisors(100)
9
sage: number_of_divisors(-720)
30

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: number_of_divisors(int8(100))
9
sage: from gmpy2 import mpz
sage: number_of_divisors(mpz(100))
9

sage.arith.misc.odd_part(n)

The odd part of the integer \(n\). This is \(n / 2^v\), where \(v = \mathrm{valuation}(n,2)\).

EXAMPLES:

sage: odd_part(5)
5
sage: odd_part(4)
1
sage: odd_part(factorial(31))
122529844256906551386796875

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: odd_part(int8(5))
5
sage: from gmpy2 import mpz
sage: odd_part(mpz(5))
5

sage.arith.misc.power_mod(a, n, m)

Return the n-th power of a modulo the integer m.

EXAMPLES:

sage: power_mod(0,0,5)
Traceback (most recent call last):
...
ArithmeticError: 0^0 is undefined.
sage: power_mod(2,390,391)
285
sage: power_mod(2,-1,7)
4
sage: power_mod(11,1,7)
4
sage: R.<x> = ZZ[]
sage: power_mod(3*x, 10, 7)
4*x^10

sage: power_mod(11,1,0)
Traceback (most recent call last):
...
ZeroDivisionError: modulus must be nonzero.

Tests with numpy and gmpy2 numbers:

sage: from numpy import int32
sage: power_mod(int32(2),int32(390),int32(391))
285
sage: from gmpy2 import mpz
sage: power_mod(mpz(2),mpz(390),mpz(391))
mpz(285)

sage.arith.misc.previous_prime(n)

The largest prime < n. The result is provably correct. If n <= 1, this function raises a ValueError.

EXAMPLES:

sage: previous_prime(10)
7
sage: previous_prime(7)
5
sage: previous_prime(8)
7
sage: previous_prime(7)
5
sage: previous_prime(5)
3
sage: previous_prime(3)
2
sage: previous_prime(2)
Traceback (most recent call last):
...
ValueError: no previous prime
sage: previous_prime(1)
Traceback (most recent call last):
...
ValueError: no previous prime
sage: previous_prime(-20)
Traceback (most recent call last):
...
ValueError: no previous prime

sage.arith.misc.previous_prime_power(n)

Return the largest prime power smaller than n.

The result is provably correct. If n is smaller or equal than 2 this function raises an error.

This function simply call the method Integer.previous_prime_power() of Integers.

See also

• is_prime_power() (and Integer.is_prime_power())
• next_prime_power() (and Integer.next_prime_power())

EXAMPLES:

sage: previous_prime_power(3)
2
sage: previous_prime_power(10)
9
sage: previous_prime_power(7)
5
sage: previous_prime_power(127)
125

The same results can be obtained with:

sage: 3.previous_prime_power()
2
sage: 10.previous_prime_power()
9
sage: 7.previous_prime_power()
5
sage: 127.previous_prime_power()
125

Input less than or equal to \(2\) raises errors:

sage: previous_prime_power(2)
Traceback (most recent call last):
...
ValueError: no prime power less than 2
sage: previous_prime_power(-10)
Traceback (most recent call last):
...
ValueError: no prime power less than 2

sage: n = previous_prime_power(2^16 - 1)
sage: while is_prime(n):
....:     n = previous_prime_power(n)
sage: factor(n)
251^2

sage.arith.misc.prime_divisors(n)

Return the list of prime divisors (up to units) of this element of a unique factorization domain.

INPUT:

• n – any object which can be decomposed into prime factors

OUTPUT:

A list of prime factors of n. For integers, this list is sorted in increasing order.

EXAMPLES:

Prime divisors of positive integers:

sage: prime_divisors(1)
[]
sage: prime_divisors(100)
[2, 5]
sage: prime_divisors(2004)
[2, 3, 167]

If n is negative, we do not include -1 among the prime divisors, since -1 is not a prime number:

sage: prime_divisors(-100)
[2, 5]

For polynomials we get all irreducible factors:

sage: R.<x> = PolynomialRing(QQ)
sage: prime_divisors(x^12 - 1)
[x - 1, x + 1, x^2 - x + 1, x^2 + 1, x^2 + x + 1, x^4 - x^2 + 1]

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: prime_divisors(int8(-100))
[2, 5]
sage: from gmpy2 import mpz
sage: prime_divisors(mpz(-100))
[2, 5]

sage.arith.misc.prime_factors(n)

Return the list of prime divisors (up to units) of this element of a unique factorization domain.

INPUT:

• n – any object which can be decomposed into prime factors

OUTPUT:

A list of prime factors of n. For integers, this list is sorted in increasing order.

EXAMPLES:

Prime divisors of positive integers:

sage: prime_divisors(1)
[]
sage: prime_divisors(100)
[2, 5]
sage: prime_divisors(2004)
[2, 3, 167]

If n is negative, we do not include -1 among the prime divisors, since -1 is not a prime number:

sage: prime_divisors(-100)
[2, 5]

For polynomials we get all irreducible factors:

sage: R.<x> = PolynomialRing(QQ)
sage: prime_divisors(x^12 - 1)
[x - 1, x + 1, x^2 - x + 1, x^2 + 1, x^2 + x + 1, x^4 - x^2 + 1]

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: prime_divisors(int8(-100))
[2, 5]
sage: from gmpy2 import mpz
sage: prime_divisors(mpz(-100))
[2, 5]

sage.arith.misc.prime_powers(start, stop=None)

List of all positive primes powers between start and stop-1, inclusive. If the second argument is omitted, returns the prime powers up to the first argument.

INPUT:

• start - an integer. If two inputs are given, a lower bound for the returned set of prime powers. If this is the only input, then it is an upper bound.
• stop - an integer (default: None). An upper bound for the returned set of prime powers.

OUTPUT:

The set of all prime powers between start and stop or, if only one argument is passed, the set of all prime powers between 1 and start. The number \(n\) is a prime power if \(n=p^k\), where \(p\) is a prime number and \(k\) is a positive integer. Thus, \(1\) is not a prime power.

EXAMPLES:

sage: prime_powers(20)
[2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19]
sage: len(prime_powers(1000))
193
sage: len(prime_range(1000))
168

sage: a = [z for z in range(95,1234) if is_prime_power(z)]
sage: b = prime_powers(95,1234)
sage: len(b)
194
sage: len(a)
194
sage: a[:10]
[97, 101, 103, 107, 109, 113, 121, 125, 127, 128]
sage: b[:10]
[97, 101, 103, 107, 109, 113, 121, 125, 127, 128]
sage: a == b
True

sage: prime_powers(100) == [i for i in range(100) if is_prime_power(i)]
True

sage: prime_powers(10,7)
[]
sage: prime_powers(-5)
[]
sage: prime_powers(-1,3)
[2]

sage.arith.misc.prime_to_m_part(n, m)

Return the prime-to-m part of n.

This is the largest divisor of n that is coprime to m.

INPUT:

• n – Integer (nonzero)
• m – Integer

OUTPUT: Integer

EXAMPLES:

sage: prime_to_m_part(240,2)
15
sage: prime_to_m_part(240,3)
80
sage: prime_to_m_part(240,5)
48
sage: prime_to_m_part(43434,20)
21717

Note that integers also have a method with the same name:

sage: 240.prime_to_m_part(2)
15

Tests with numpy and gmpy2 numbers:

sage: from numpy import int16
sage: prime_to_m_part(int16(240), int16(2))
15
sage: from gmpy2 import mpz
sage: prime_to_m_part(mpz(240), mpz(2))
15

sage.arith.misc.primes(start, stop=None, proof=None)

Return an iterator over all primes between start and stop-1, inclusive. This is much slower than prime_range, but potentially uses less memory. As with next_prime(), the optional argument proof controls whether the numbers returned are guaranteed to be prime or not.

This command is like the Python 2 xrange command, except it only iterates over primes. In some cases it is better to use primes than prime_range, because primes does not build a list of all primes in the range in memory all at once. However, it is potentially much slower since it simply calls the next_prime() function repeatedly, and next_prime() is slow.

INPUT:

• start - an integer - lower bound for the primes
• stop - an integer (or infinity) optional argument - giving upper (open) bound for the primes
• proof - bool or None (default: None) If True, the function yields only proven primes. If False, the function uses a pseudo-primality test, which is much faster for really big numbers but does not provide a proof of primality. If None, uses the global default (see sage.structure.proof.proof)

OUTPUT:

• an iterator over primes from start to stop-1, inclusive

EXAMPLES:

sage: for p in primes(5,10):
....:     print(p)
5
7
sage: list(primes(13))
[2, 3, 5, 7, 11]
sage: list(primes(10000000000, 10000000100))
[10000000019, 10000000033, 10000000061, 10000000069, 10000000097]
sage: max(primes(10^100, 10^100+10^4, proof=False))
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000009631
sage: next(p for p in primes(10^20, infinity) if is_prime(2*p+1))
100000000000000001243

sage.arith.misc.primes_first_n(n, leave_pari=False)

Return the first \(n\) primes.

INPUT:

• \(n\) - a nonnegative integer

OUTPUT:

• a list of the first \(n\) prime numbers.

EXAMPLES:

sage: primes_first_n(10)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
sage: len(primes_first_n(1000))
1000
sage: primes_first_n(0)
[]

sage.arith.misc.primitive_root(n, check=True)

Return a positive integer that generates the multiplicative group of integers modulo \(n\), if one exists; otherwise, raise a ValueError.

A primitive root exists if \(n=4\) or \(n=p^k\) or \(n=2p^k\), where \(p\) is an odd prime and \(k\) is a nonnegative number.

INPUT:

• n – a non-zero integer
• check – bool (default: True); if False, then \(n\) is assumed to be a positive integer possessing a primitive root, and behavior is undefined otherwise.

OUTPUT:

A primitive root of \(n\). If \(n\) is prime, this is the smallest primitive root.

EXAMPLES:

sage: primitive_root(23)
5
sage: primitive_root(-46)
5
sage: primitive_root(25)
2
sage: print([primitive_root(p) for p in primes(100)])
[1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5]
sage: primitive_root(8)
Traceback (most recent call last):
...
ValueError: no primitive root

Note

It takes extra work to check if \(n\) has a primitive root; to avoid this, use check=False, which may slightly speed things up (but could also result in undefined behavior). For example, the second call below is an order of magnitude faster than the first:

sage: n = 10^50 + 151   # a prime
sage: primitive_root(n)
11
sage: primitive_root(n, check=False)
11

sage.arith.misc.quadratic_residues(n)

Return a sorted list of all squares modulo the integer \(n\) in the range \(0\leq x < |n|\).

EXAMPLES:

sage: quadratic_residues(11)
[0, 1, 3, 4, 5, 9]
sage: quadratic_residues(1)
[0]
sage: quadratic_residues(2)
[0, 1]
sage: quadratic_residues(8)
[0, 1, 4]
sage: quadratic_residues(-10)
[0, 1, 4, 5, 6, 9]
sage: v = quadratic_residues(1000); len(v)
159

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: quadratic_residues(int8(11))
[0, 1, 3, 4, 5, 9]
sage: from gmpy2 import mpz
sage: quadratic_residues(mpz(11))
[0, 1, 3, 4, 5, 9]

sage.arith.misc.radical(n, *args, **kwds)

Return the product of the prime divisors of n.

This calls n.radical(*args, **kwds).

EXAMPLES:

sage: radical(2 * 3^2 * 5^5)
30
sage: radical(0)
Traceback (most recent call last):
...
ArithmeticError: Radical of 0 not defined.
sage: K.<i> = QuadraticField(-1)
sage: radical(K(2))
i + 1

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: radical(int8(50))
10
sage: from gmpy2 import mpz
sage: radical(mpz(50))
10

sage.arith.misc.random_prime(n, proof=None, lbound=2)

Return a random prime \(p\) between lbound and \(n\).

The returned prime \(p\) satisfies lbound \(\leq p \leq n\).

The returned prime \(p\) is chosen uniformly at random from the set of prime numbers less than or equal to \(n\).

INPUT:

• n - an integer >= 2.
• proof - bool or None (default: None) If False, the function uses a pseudo-primality test, which is much faster for really big numbers but does not provide a proof of primality. If None, uses the global default (see sage.structure.proof.proof)
• lbound - an integer >= 2, lower bound for the chosen primes

EXAMPLES:

sage: random_prime(100000)
30029
sage: random_prime(2)
2

Here we generate a random prime between 100 and 200:

sage: random_prime(200, lbound=100)
167

If all we care about is finding a pseudo prime, then we can pass in proof=False

sage: random_prime(200, proof=False, lbound=100)
197

AUTHORS:

• Jon Hanke (2006-08-08): with standard Stein cleanup
• Jonathan Bober (2007-03-17)

sage.arith.misc.rational_reconstruction(a, m, algorithm='fast')

This function tries to compute \(x/y\), where \(x/y\) is a rational number in lowest terms such that the reduction of \(x/y\) modulo \(m\) is equal to \(a\) and the absolute values of \(x\) and \(y\) are both \(\le \sqrt{m/2}\). If such \(x/y\) exists, that pair is unique and this function returns it. If no such pair exists, this function raises ZeroDivisionError.

An efficient algorithm for computing rational reconstruction is very similar to the extended Euclidean algorithm. For more details, see Knuth, Vol 2, 3rd ed, pages 656-657.

INPUT:

• a – an integer
• m – a modulus
• algorithm – (default: ‘fast’)
  • 'fast' - a fast implementation using direct MPIR calls in Cython.

OUTPUT:

Numerator and denominator \(n\), \(d\) of the unique rational number \(r=n/d\), if it exists, with \(n\) and \(|d| \le \sqrt{N/2}\). Return \((0,0)\) if no such number exists.

The algorithm for rational reconstruction is described (with a complete nontrivial proof) on pages 656-657 of Knuth, Vol 2, 3rd ed. as the solution to exercise 51 on page 379. See in particular the conclusion paragraph right in the middle of page 657, which describes the algorithm thus:

This discussion proves that the problem can be solved efficiently by applying Algorithm 4.5.2X with \(u=m\) and \(v=a\), but with the following replacement for step X2: If \(v3 \le \sqrt{m/2}\), the algorithm terminates. The pair \((x,y)=(|v2|,v3*\mathrm{sign}(v2))\) is then the unique solution, provided that \(x\) and \(y\) are coprime and \(x \le \sqrt{m/2}\); otherwise there is no solution. (Alg 4.5.2X is the extended Euclidean algorithm.)

Knuth remarks that this algorithm is due to Wang, Kornerup, and Gregory from around 1983.

EXAMPLES:

sage: m = 100000
sage: (119*inverse_mod(53,m))%m
11323
sage: rational_reconstruction(11323,m)
119/53

sage: rational_reconstruction(400,1000)
Traceback (most recent call last):
...
ArithmeticError: rational reconstruction of 400 (mod 1000) does not exist

sage: rational_reconstruction(3, 292393)
3
sage: a = Integers(292393)(45/97); a
204977
sage: rational_reconstruction(a, 292393, algorithm='fast')
45/97
sage: rational_reconstruction(293048, 292393)
Traceback (most recent call last):
...
ArithmeticError: rational reconstruction of 655 (mod 292393) does not exist
sage: rational_reconstruction(0, 0)
Traceback (most recent call last):
...
ZeroDivisionError: rational reconstruction with zero modulus
sage: rational_reconstruction(0, 1, algorithm="foobar")
Traceback (most recent call last):
...
ValueError: unknown algorithm 'foobar'

Tests with numpy and gmpy2 numbers:

sage: from numpy import int32
sage: rational_reconstruction(int32(3), int32(292393))
3
sage: from gmpy2 import mpz
sage: rational_reconstruction(mpz(3), mpz(292393))
3

sage.arith.misc.rising_factorial(x, a)

Return the rising factorial \((x)^a\).

The notation in the literature is a mess: often \((x)^a\), but there are many other notations: GKP: Concrete Mathematics uses \(x^{\overline{a}}\).

The rising factorial is also known as the Pochhammer symbol, see Maple and Mathematica.

Definition: for integer \(a \ge 0\) we have \(x(x+1) \cdots (x+a-1)\). In all other cases we use the GAMMA-function: \(\frac {\Gamma(x+a)} {\Gamma(x)}\).

INPUT:

• x – element of a ring
• a – a non-negative integer or
• x and a – any numbers

OUTPUT: the rising factorial

See also

falling_factorial()

EXAMPLES:

sage: rising_factorial(10,3)
1320

sage: rising_factorial(10,RR('3.0'))
1320.00000000000

sage: rising_factorial(10,RR('3.3'))
2826.38895824964

sage: a = rising_factorial(1+I, I); a
gamma(2*I + 1)/gamma(I + 1)
sage: CC(a)
0.266816390637832 + 0.122783354006372*I

sage: a = rising_factorial(I, 4); a
-10

sage: x = polygen(ZZ)
sage: rising_factorial(x, 4)
x^4 + 6*x^3 + 11*x^2 + 6*x

AUTHORS:

• Jaap Spies (2006-03-05)

sage.arith.misc.sort_complex_numbers_for_display(nums)

Given a list of complex numbers (or a list of tuples, where the first element of each tuple is a complex number), we sort the list in a “pretty” order. First come the real numbers (with zero imaginary part), then the complex numbers sorted according to their real part. If two complex numbers have the same real part, then they are sorted according to their imaginary part.

This is not a useful function mathematically (not least because there is no principled way to determine whether the real components should be treated as equal or not). It is called by various polynomial root-finders; its purpose is to make doctest printing more reproducible.

We deliberately choose a cumbersome name for this function to discourage use, since it is mathematically meaningless.

EXAMPLES:

sage: import sage.arith.misc
sage: sort_c = sort_complex_numbers_for_display
sage: nums = [CDF(i) for i in range(3)]
sage: for i in range(3):
....:     nums.append(CDF(i + RDF.random_element(-3e-11, 3e-11),
....:                     RDF.random_element()))
....:     nums.append(CDF(i + RDF.random_element(-3e-11, 3e-11),
....:                     RDF.random_element()))
sage: shuffle(nums)
sage: sort_c(nums)
[0.0, 1.0, 2.0, -2.862406201002009e-11 - 0.7088740263015161*I, 2.2108362706985576e-11 - 0.43681052967509904*I, 1.0000000000138833 - 0.7587654737635712*I, 0.9999999999760288 - 0.7238965893336062*I, 1.9999999999874383 - 0.4560801012073723*I, 1.9999999999869107 + 0.6090836283134269*I]

sage.arith.misc.squarefree_divisors(x)

Return an iterator over the squarefree divisors (up to units) of this ring element.

Depends on the output of the prime_divisors function.

Squarefree divisors of an integer are not necessarily yielded in increasing order.

INPUT:

• x – an element of any ring for which the prime_divisors function works.

EXAMPLES:

Integers with few prime divisors:

sage: list(squarefree_divisors(7))
[1, 7]
sage: list(squarefree_divisors(6))
[1, 2, 3, 6]
sage: list(squarefree_divisors(12))
[1, 2, 3, 6]

Squarefree divisors are not yielded in increasing order:

sage: list(squarefree_divisors(30))
[1, 2, 3, 6, 5, 10, 15, 30]

sage.arith.misc.subfactorial(n)

Subfactorial or rencontres numbers, or derangements: number of permutations of \(n\) elements with no fixed points.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

sage: subfactorial(0)
1
sage: subfactorial(1)
0
sage: subfactorial(8)
14833

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: subfactorial(int8(8))
14833
sage: from gmpy2 import mpz
sage: subfactorial(mpz(8))
14833

AUTHORS:

• Jaap Spies (2007-01-23)

sage.arith.misc.sum_of_k_squares(k, n)

Write the integer \(n\) as a sum of \(k\) integer squares if possible; otherwise raise a ValueError.

INPUT:

• k – a non-negative integer
• n – an integer

OUTPUT: a tuple \((x_1, ..., x_k)\) of non-negative integers such that their squares sum to \(n\).

EXAMPLES:

sage: sum_of_k_squares(2, 9634)
(15, 97)
sage: sum_of_k_squares(3, 9634)
(0, 15, 97)
sage: sum_of_k_squares(4, 9634)
(1, 2, 5, 98)
sage: sum_of_k_squares(5, 9634)
(0, 1, 2, 5, 98)
sage: sum_of_k_squares(6, 11^1111-1)
(19215400822645944253860920437586326284, 37204645194585992174252915693267578306, 3473654819477394665857484221256136567800161086815834297092488779216863122, 5860191799617673633547572610351797996721850737768032876360978911074629287841061578270832330322236796556721252602860754789786937515870682024273948, 20457423294558182494001919812379023992538802203730791019728543439765347851316366537094696896669915675685581905102118246887673397020172285247862426612188418787649371716686651256443143210952163970564228423098202682066311189439731080552623884051737264415984619097656479060977602722566383385989, 311628095411678159849237738619458396497534696043580912225334269371611836910345930320700816649653412141574887113710604828156159177769285115652741014638785285820578943010943846225597311231847997461959204894255074229895666356909071243390280307709880906261008237873840245959883405303580405277298513108957483306488193844321589356441983980532251051786704380984788999660195252373574924026139168936921591652831237741973242604363696352878914129671292072201700073286987126265965322808664802662993006926302359371379531571194266134916767573373504566621665949840469229781956838744551367172353)
sage: sum_of_k_squares(7, 0)
(0, 0, 0, 0, 0, 0, 0)
sage: sum_of_k_squares(30,999999)
(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 7, 44, 999)
sage: sum_of_k_squares(1, 9)
(3,)
sage: sum_of_k_squares(1, 10)
Traceback (most recent call last):
...
ValueError: 10 is not a sum of 1 square
sage: sum_of_k_squares(1, -10)
Traceback (most recent call last):
...
ValueError: -10 is not a sum of 1 square
sage: sum_of_k_squares(0, 9)
Traceback (most recent call last):
...
ValueError: 9 is not a sum of 0 squares
sage: sum_of_k_squares(0, 0)
()
sage: sum_of_k_squares(7, -1)
Traceback (most recent call last):
...
ValueError: -1 is not a sum of 7 squares
sage: sum_of_k_squares(-1, 0)
Traceback (most recent call last):
...
ValueError: k = -1 must be non-negative

Tests with numpy and gmpy2 numbers:

sage: from numpy import int16
sage: sum_of_k_squares(int16(2), int16(9634))
(15, 97)
sage: from gmpy2 import mpz
sage: sum_of_k_squares(mpz(2), mpz(9634))
(15, 97)

sage.arith.misc.three_squares(n)

Write the integer \(n\) as a sum of three integer squares if possible; otherwise raise a ValueError.

INPUT:

• n – an integer

OUTPUT: a tuple \((a,b,c)\) of non-negative integers such that \(n = a^2 + b^2 + c^2\) with \(a <= b <= c\).

EXAMPLES:

sage: three_squares(389)
(1, 8, 18)
sage: three_squares(946)
(9, 9, 28)
sage: three_squares(2986)
(3, 24, 49)
sage: three_squares(7^100)
(0, 0, 1798465042647412146620280340569649349251249)
sage: three_squares(11^111-1)
(616274160655975340150706442680, 901582938385735143295060746161, 6270382387635744140394001363065311967964099981788593947233)
sage: three_squares(7 * 2^41)
(1048576, 2097152, 3145728)
sage: three_squares(7 * 2^42)
Traceback (most recent call last):
...
ValueError: 30786325577728 is not a sum of 3 squares
sage: three_squares(0)
(0, 0, 0)
sage: three_squares(-1)
Traceback (most recent call last):
...
ValueError: -1 is not a sum of 3 squares

ALGORITHM:

See http://www.schorn.ch/howto.html

sage.arith.misc.trial_division(n, bound=None)

Return the smallest prime divisor <= bound of the positive integer n, or n if there is no such prime. If the optional argument bound is omitted, then bound <= n.

INPUT:

• n - a positive integer
• bound - (optional) a positive integer

OUTPUT:

• int - a prime p=bound that divides n, or n if there is no such prime.

EXAMPLES:

sage: trial_division(15)
3
sage: trial_division(91)
7
sage: trial_division(11)
11
sage: trial_division(387833, 300)
387833
sage: # 300 is not big enough to split off a
sage: # factor, but 400 is.
sage: trial_division(387833, 400)
389

Tests with numpy and gmpy2 numbers:

sage: from numpy import int8
sage: trial_division(int8(91))
7
sage: from gmpy2 import mpz
sage: trial_division(mpz(91))
7

sage.arith.misc.two_squares(n)

Write the integer \(n\) as a sum of two integer squares if possible; otherwise raise a ValueError.

INPUT:

• n – an integer

OUTPUT: a tuple \((a,b)\) of non-negative integers such that \(n = a^2 + b^2\) with \(a <= b\).

EXAMPLES:

sage: two_squares(389)
(10, 17)
sage: two_squares(21)
Traceback (most recent call last):
...
ValueError: 21 is not a sum of 2 squares
sage: two_squares(21^2)
(0, 21)
sage: a,b = two_squares(100000000000000000129); a,b
(4418521500, 8970878873)
sage: a^2 + b^2
100000000000000000129
sage: two_squares(2^222+1)
(253801659504708621991421712450521, 2583712713213354898490304645018692)
sage: two_squares(0)
(0, 0)
sage: two_squares(-1)
Traceback (most recent call last):
...
ValueError: -1 is not a sum of 2 squares

ALGORITHM:

See http://www.schorn.ch/howto.html

sage.arith.misc.valuation(m, *args, **kwds)

Return the valuation of m.

This function simply calls the m.valuation() method. See the documentation of m.valuation() for a more precise description.

Note that the use of this functions is discouraged as it is better to use m.valuation() directly.

Note

This is not always a valuation in the mathematical sense. For more information see: sage.rings.finite_rings.integer_mod.IntegerMod_int.valuation

EXAMPLES:

sage: valuation(512,2)
9
sage: valuation(1,2)
0
sage: valuation(5/9, 3)
-2

Valuation of 0 is defined, but valuation with respect to 0 is not:

sage: valuation(0,7)
+Infinity
sage: valuation(3,0)
Traceback (most recent call last):
...
ValueError: You can only compute the valuation with respect to a integer larger than 1.

Here are some other examples:

sage: valuation(100,10)
2
sage: valuation(200,10)
2
sage: valuation(243,3)
5
sage: valuation(243*10007,3)
5
sage: valuation(243*10007,10007)
1
sage: y = QQ['y'].gen()
sage: valuation(y^3, y)
3
sage: x = QQ[['x']].gen()
sage: valuation((x^3-x^2)/(x-4))
2
sage: valuation(4r,2r)
2
sage: valuation(1r,1r)
Traceback (most recent call last):
...
ValueError: You can only compute the valuation with respect to a integer larger than 1.
sage: from numpy import int16
sage: valuation(int16(512), int16(2))
9
sage: from gmpy2 import mpz
sage: valuation(mpz(512), mpz(2))
9

sage.arith.misc.xgcd(a, b)

Return a triple (g,s,t) such that \(g = s\cdot a+t\cdot b = \gcd(a,b)\).

Note

One exception is if \(a\) and \(b\) are not in a principal ideal domain (see Wikipedia article Principal_ideal_domain), e.g., they are both polynomials over the integers. Then this function can’t in general return (g,s,t) as above, since they need not exist. Instead, over the integers, we first multiply \(g\) by a divisor of the resultant of \(a/g\) and \(b/g\), up to sign.

INPUT:

• a, b - integers or more generally, element of a ring for which the xgcd make sense (e.g. a field or univariate polynomials).

OUTPUT:

• g, s, t - such that \(g = s\cdot a + t\cdot b\)

Note

There is no guarantee that the returned cofactors (s and t) are minimal.

EXAMPLES:

sage: xgcd(56, 44)
(4, 4, -5)
sage: 4*56 + (-5)*44
4

sage: g, a, b = xgcd(5/1, 7/1); g, a, b
(1, 3, -2)
sage: a*(5/1) + b*(7/1) == g
True

sage: x = polygen(QQ)
sage: xgcd(x^3 - 1, x^2 - 1)
(x - 1, 1, -x)

sage: K.<g> = NumberField(x^2-3)
sage: g.xgcd(g+2)
(1, 1/3*g, 0)

sage: R.<a,b> = K[]
sage: S.<y> = R.fraction_field()[]
sage: xgcd(y^2, a*y+b)
(1, a^2/b^2, ((-a)/b^2)*y + 1/b)
sage: xgcd((b+g)*y^2, (a+g)*y+b)
(1, (a^2 + (2*g)*a + 3)/(b^3 + (g)*b^2), ((-a + (-g))/b^2)*y + 1/b)

Here is an example of a xgcd for two polynomials over the integers, where the linear combination is not the gcd but the gcd multiplied by the resultant:

sage: R.<x> = ZZ[]
sage: gcd(2*x*(x-1), x^2)
x
sage: xgcd(2*x*(x-1), x^2)
(2*x, -1, 2)
sage: (2*(x-1)).resultant(x)
2

Tests with numpy and gmpy2 types:

sage: from numpy import int8
sage: xgcd(4,int8(8))
(4, 1, 0)
sage: xgcd(int8(4),int8(8))
(4, 1, 0)
sage: from gmpy2 import mpz
sage: xgcd(mpz(4), mpz(8))
(4, 1, 0)
sage: xgcd(4, mpz(8))
(4, 1, 0)

sage.arith.misc.xkcd(n='')

This function is similar to the xgcd function, but behaves in a completely different way.

INPUT:

• n – an integer (optional)

OUTPUT: a fragment of HTML

EXAMPLES:

sage: xkcd(353)  # optional - internet
<h1>Python</h1><img src="https://imgs.xkcd.com/comics/python.png" title="I wrote 20 short programs in Python yesterday.  It was wonderful.  Perl, I'm leaving you."><div>Source: <a href="http://xkcd.com/353" target="_blank">http://xkcd.com/353</a></div>

sage.arith.misc.xlcm(m, n)

Extended lcm function: given two positive integers \(m,n\), returns a triple \((l,m_1,n_1)\) such that \(l=\mathop{\mathrm{lcm}}(m,n)=m_1 \cdot n_1\) where \(m_1|m\), \(n_1|n\) and \(\gcd(m_1,n_1)=1\), all with no factorization.

Used to construct an element of order \(l\) from elements of orders \(m,n\) in any group: see sage/groups/generic.py for examples.

EXAMPLES:

sage: xlcm(120,36)
(360, 40, 9)