Elementary number theory - Constructions

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# Elementary number theory

## Taking modular powers

How do I compute modular powers in Sage?

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

Sage

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

Python

    >>> 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

    sage: 51.powermod(99203843984,97)
    96

[Python]

    >>> 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

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

Python

    >>> 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

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

[Python]

    >>> 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

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

Python

    >>> 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

    sage: next_prime(2005)
          2011

[Python]

    >>> 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

    sage: pari(10).primepi()
          4

Python

    >>> 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

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

[Python]

    >>> 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

    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

Python

    >>> 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

    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]

Python

    >>> 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

    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]

[Python]

    >>> 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]