************************
Elementary number theory
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Taking modular powers
=====================
How do I compute modular powers in Sage?
To compute :math:`51^{2006} \pmod{97}` in Sage, type
::
sage: R = Integers(97)
sage: a = R(51)
sage: a^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
.. index:: discrete logs
Discrete logs
=============
To find a number :math:`x` such that
:math:`b^x\equiv a \pmod m` (the discrete log of
:math:`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
This also works over finite fields:
::
sage: FF = FiniteField(16,"a")
sage: a = FF.gen()
sage: c = a^7
sage: 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
The usage of ``next_prime`` is self-explanatory:
::
sage: next_prime(2005)
2011
The Pari command ``primepi`` is used via the command
``pari(x).primepi()``. This returns the number of primes
:math:`\leq x`, for example:
::
sage: pari(10).primepi()
4
Using ``primes_first_n`` or ``primes`` one can check that, indeed,
there are :math:`4` primes up to :math:`10`:
::
sage: primes_first_n(5)
[2, 3, 5, 7, 11]
sage: list(primes(1, 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 number (usually denoted
:math:`d(n)`) of divisors of :math:`n` and ``sigma(n,k)`` for the
sum of the :math:`k^{th}` powers of the divisors of :math:`n` (so
``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
.. index:: quadratic residues
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]
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]