# math::numtheory(n) 1.1.3 tcllib "Tcl Math Library"

## Name

math::numtheory - Number Theory

## Description

This package is for collecting various number-theoretic operations, with a slight bias to prime numbers.

math::numtheory::isprime N ?option value ...?

The isprime command tests whether the integer N is a prime, returning a boolean true value for prime N and a boolean false value for non-prime N. The formal definition of 'prime' used is the conventional, that the number being tested is greater than 1 and only has trivial divisors.

To be precise, the return value is one of 0 (if N is definitely not a prime), 1 (if N is definitely a prime), and on (if N is probably prime); the latter two are both boolean true values. The case that an integer may be classified as "probably prime" arises because the Miller-Rabin algorithm used in the test implementation is basically probabilistic, and may if we are unlucky fail to detect that a number is in fact composite. Options may be used to select the risk of such "false positives" in the test. 1 is returned for "small" N (which currently means N < 118670087467), where it is known that no false positives are possible.

The only option currently defined is:

-randommr repetitions

which controls how many times the Miller-Rabin test should be repeated with randomly chosen bases. Each repetition reduces the probability of a false positive by a factor at least 4. The default for repetitions is 4.

Unknown options are silently ignored.

math::numtheory::firstNprimes N

Return the first N primes

integer N (in)

Number of primes to return

math::numtheory::primesLowerThan N

Return the prime numbers lower/equal to N

integer N (in)

Maximum number to consider

math::numtheory::primeFactors N

Return a list of the prime numbers in the number N

integer N (in)

Number to be factorised

math::numtheory::primesLowerThan N

Return the prime numbers lower/equal to N

integer N (in)

Maximum number to consider

math::numtheory::primeFactors N

Return a list of the prime numbers in the number N

integer N (in)

Number to be factorised

math::numtheory::uniquePrimeFactors N

Return a list of the unique prime numbers in the number N

integer N (in)

Number to be factorised

math::numtheory::factors N

Return a list of all unique factors in the number N, including 1 and N itself

integer N (in)

Number to be factorised

math::numtheory::totient N

Evaluate the Euler totient function for the number N (number of numbers relatively prime to N)

integer N (in)

Number in question

math::numtheory::moebius N

Evaluate the Moebius function for the number N

integer N (in)

Number in question

math::numtheory::legendre a p

Evaluate the Legendre symbol (a/p)

integer a (in)

Upper number in the symbol

integer p (in)

Lower number in the symbol (must be non-zero)

math::numtheory::jacobi a b

Evaluate the Jacobi symbol (a/b)

integer a (in)

Upper number in the symbol

integer b (in)

Lower number in the symbol (must be odd)

math::numtheory::gcd m n

Return the greatest common divisor of m and n

integer m (in)

First number

integer n (in)

Second number

math::numtheory::lcm m n

Return the lowest common multiple of m and n

integer m (in)

First number

integer n (in)

Second number

math::numtheory::numberPrimesGauss N

Estimate the number of primes according the formula by Gauss.

integer N (in)

Number in question, should be larger than 0

math::numtheory::numberPrimesLegendre N

Estimate the number of primes according the formula by Legendre.

integer N (in)

Number in question, should be larger than 0

math::numtheory::numberPrimesLegendreModified N

Estimate the number of primes according the modified formula by Legendre.

integer N (in)

Number in question, should be larger than 0

math::numtheory::differenceNumberPrimesLegendreModified lower upper

Estimate the number of primes between tow limits according the modified formula by Legendre.

integer lower (in)

Lower limit for the primes, should be larger than 0

integer upper (in)

Upper limit for the primes, should be larger than 0

math::numtheory::listPrimePairs lower upper step

Return a list of pairs of primes each differing by the given step.

integer lower (in)

Lower limit for the primes, should be larger than 0

integer upper (in)

Upper limit for the primes, should be larger than the lower limit

integer step (in)

Step by which the primes should differ, defaults to 2

math::numtheory::listPrimeProgressions lower upper step

Return a list of lists of primes each differing by the given step from the previous one.

integer lower (in)

Lower limit for the primes, should be larger than 0

integer upper (in)

Upper limit for the primes, should be larger than the lower limit

integer step (in)

Step by which the primes should differ, defaults to 2

## Bugs, Ideas, Feedback

This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category math :: numtheory of the Tcllib Trackers. Please also report any ideas for enhancements you may have for either package and/or documentation.

When proposing code changes, please provide unified diffs, i.e the output of diff -u.

Note further that attachments are strongly preferred over inlined patches. Attachments can be made by going to the Edit form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar.

Mathematics