# numtheory - Man Page

Number Theory

## Synopsis

`package require `

**Tcl ?8.5?**

`package require `

**math::numtheory ?1.0?**

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

**math::numtheory::firstNprimes** *N*

**math::numtheory::primesLowerThan** *N*

**math::numtheory::primeFactors** *N*

**math::numtheory::uniquePrimeFactors** *N*

**math::numtheory::factors** *N*

**math::numtheory::totient** *N*

**math::numtheory::moebius** *N*

**math::numtheory::legendre** *a p*

**math::numtheory::jacobi** *a b*

**math::numtheory::gcd** *m n*

**math::numtheory::lcm** *m n*

**math::numtheory::numberPrimesGauss** *N*

**math::numtheory::numberPrimesLegendre** *N*

**math::numtheory::numberPrimesLegendreModified** *N*

## 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

- integer
**math::numtheory::primesLowerThan***N*Return the prime numbers lower/equal to N

- integer
*N*(in) Maximum number to consider

- integer
**math::numtheory::primeFactors***N*Return a list of the prime numbers in the number N

- integer
*N*(in) Number to be factorised

- integer
**math::numtheory::uniquePrimeFactors***N*Return a list of the

*unique*prime numbers in the number N- integer
*N*(in) Number to be factorised

- integer
**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

- integer
**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

- integer
**math::numtheory::moebius***N*Evaluate the Moebius function for the number N

- integer
*N*(in) Number in question

- integer
**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)

- integer
**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)

- integer
**math::numtheory::gcd***m n*Return the greatest common divisor of

*m*and*n*- integer
*m*(in) First number

- integer
*n*(in) Second number

- integer
**math::numtheory::lcm***m n*Return the lowest common multiple of

*m*and*n*- integer
*m*(in) First number

- integer
*n*(in) Second number

- integer
**math::numtheory::numberPrimesGauss***N*Estimate the number of primes according the formula by Gauss.

- integer
*N*(in) Number in question

- integer
**math::numtheory::numberPrimesLegendre***N*Estimate the number of primes according the formula by Legendre.

- integer
*N*(in) Number in question

- integer
**math::numtheory::numberPrimesLegendreModified***N*Estimate the number of primes according the modified formula by Legendre.

- integer
*N*(in) Number in question

- integer

## 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* [http://core.tcl.tk/tcllib/reportlist]. 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.

## Keywords

number theory, prime

## Category

Mathematics

## Copyright

Copyright (c) 2010 Lars Hellström <Lars dot Hellstrom at residenset dot net>