# 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


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


## 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


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 $$\leq x$$, for example:

sage: pari(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]


## Divisors¶

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

Sage uses divisors(n) for the number (usually denoted $$d(n)$$) of divisors of $$n$$ and sigma(n,k) for the sum of the $$k^{th}$$ powers of the divisors of $$n$$ (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


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]