Elementary number theory

Taking modular powers

How do I compute modular powers in Sage?

To compute \(51^{2006} \pmod{97}\) in Sage, type

sage: R = Integers(97)
sage: a = R(51)
sage: a^2006
12

>>> from sage.all import *
>>> R = Integers(Integer(97))
>>> a = R(Integer(51))
>>> a**Integer(2006)
12

Instead of R = Integers(97) you can also type R = IntegerModRing(97). Another option is to use the interface with GMP:

sage: 51.powermod(99203843984,97)
96

>>> from sage.all import *
>>> Integer(51).powermod(Integer(99203843984),Integer(97))
96

Discrete logs

To find a number \(x\) such that \(b^x\equiv a \pmod m\) (the discrete log of \(a \pmod m\)), you can call ‘s log command:

sage: r = Integers(125)
sage: b = r.multiplicative_generator()^3
sage: a = b^17
sage: a.log(b)
17

>>> from sage.all import *
>>> r = Integers(Integer(125))
>>> b = r.multiplicative_generator()**Integer(3)
>>> a = b**Integer(17)
>>> a.log(b)
17

This also works over finite fields:

sage: FF = FiniteField(16,"a")
sage: a = FF.gen()
sage: c = a^7
sage: c.log(a)
7

>>> from sage.all import *
>>> FF = FiniteField(Integer(16),"a")
>>> a = FF.gen()
>>> c = a**Integer(7)
>>> c.log(a)
7

Prime numbers

How do you construct prime numbers in Sage?

The class Primes allows for primality testing:

sage: 2^(2^12)+1 in Primes()
False
sage: 11 in Primes()
True

>>> from sage.all import *
>>> Integer(2)**(Integer(2)**Integer(12))+Integer(1) in Primes()
False
>>> Integer(11) in Primes()
True

The usage of next_prime is self-explanatory:

sage: next_prime(2005)
      2011

>>> from sage.all import *
>>> next_prime(Integer(2005))
      2011

The Pari command primepi is used via the command pari(x).primepi(). This returns the number of primes \(\leq x\), for example:

sage: pari(10).primepi()
      4

>>> from sage.all import *
>>> pari(Integer(10)).primepi()
      4

Using primes_first_n or primes one can check that, indeed, there are \(4\) primes up to \(10\):

sage: primes_first_n(5)
[2, 3, 5, 7, 11]
sage: list(primes(1, 10))
[2, 3, 5, 7]

>>> from sage.all import *
>>> primes_first_n(Integer(5))
[2, 3, 5, 7, 11]
>>> list(primes(Integer(1), Integer(10)))
[2, 3, 5, 7]

Divisors

How do you compute the sum of the divisors of an integer in Sage?

Sage uses divisors(n) for the list of divisors of \(n\), number_of_divisors(n) for the number of divisors of \(n\) and sigma(n,k) for the sum of the \(k\)-th powers of the divisors of \(n\) (so number_of_divisors(n) and sigma(n,0) are the same).

For example:

sage: divisors(28); sum(divisors(28)); 2*28
[1, 2, 4, 7, 14, 28]
56
56
sage: sigma(28,0); sigma(28,1); sigma(28,2)
6
56
1050

>>> from sage.all import *
>>> divisors(Integer(28)); sum(divisors(Integer(28))); Integer(2)*Integer(28)
[1, 2, 4, 7, 14, 28]
56
56
>>> sigma(Integer(28),Integer(0)); sigma(Integer(28),Integer(1)); sigma(Integer(28),Integer(2))
6
56
1050

Quadratic residues

Try this:

sage: Q = quadratic_residues(23); Q
[0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18]
sage: N = [x for x in range(22) if kronecker(x,23)==-1]; N
[5, 7, 10, 11, 14, 15, 17, 19, 20, 21]

>>> from sage.all import *
>>> Q = quadratic_residues(Integer(23)); Q
[0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18]
>>> N = [x for x in range(Integer(22)) if kronecker(x,Integer(23))==-Integer(1)]; N
[5, 7, 10, 11, 14, 15, 17, 19, 20, 21]

Q is the set of quadratic residues mod 23 and N is the set of non-residues.

Here is another way to construct these using the kronecker command (which is also called the “Legendre symbol”):

sage: [x for x in range(22) if kronecker(x,23)==1]
[1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18]
sage: [x for x in range(22) if kronecker(x,23)==-1]
[5, 7, 10, 11, 14, 15, 17, 19, 20, 21]

>>> from sage.all import *
>>> [x for x in range(Integer(22)) if kronecker(x,Integer(23))==Integer(1)]
[1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18]
>>> [x for x in range(Integer(22)) if kronecker(x,Integer(23))==-Integer(1)]
[5, 7, 10, 11, 14, 15, 17, 19, 20, 21]