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NAME
math::numtheory  Number Theory
Table Of Contents
SYNOPSIS
package require Tcl ?8.5 9?
package require math::numtheory ?1.1.4?
math::numtheory::isprime N ?option value ...?
math::numtheory::firstNprimes N
math::numtheory::primesLowerThan N
math::numtheory::primeFactors 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
math::numtheory::differenceNumberPrimesLegendreModified lower upper
math::numtheory::listPrimePairs lower upper step
math::numtheory::listPrimeProgressions lower upper step
DESCRIPTION
This package is for collecting various numbertheoretic 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 nonprime 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 MillerRabin 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 MillerRabin 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

Return a list of all unique factors in the number N, including 1 and N itself
integer N (in)
Number to be factorised

Evaluate the Euler totient function for the number N (number of numbers relatively prime to N)
integer N (in)
Number in question

Evaluate the Moebius function for the number N
integer N (in)
Number in question

Evaluate the Legendre symbol (a/p)
integer a (in)
Upper number in the symbol
integer p (in)
Lower number in the symbol (must be nonzero)

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)

Return the greatest common divisor of m and n
integer m (in)
First number
integer n (in)
Second number

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 leftmost button in the secondary navigation bar.
KEYWORDS
CATEGORY
Mathematics
COPYRIGHT
Copyright © 2010 Lars HellstrÃ¶m