'\"
'\" Generated from file 'numtheory\&.man' by tcllib/doctools with format 'nroff'
'\" Copyright (c) 2010 Lars Hellström <Lars dot Hellstrom at residenset dot net>
'\"
.TH "math::numtheory" n 1\&.1\&.4 tcllib "Tcl Math Library"
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.BS
.SH NAME
math::numtheory \- Number Theory
.SH SYNOPSIS
package require \fBTcl ?8\&.5 9?\fR
.sp
package require \fBmath::numtheory ?1\&.1\&.4?\fR
.sp
\fBmath::numtheory::isprime\fR \fIN\fR ?\fIoption\fR \fIvalue\fR \&.\&.\&.?
.sp
\fBmath::numtheory::firstNprimes\fR \fIN\fR
.sp
\fBmath::numtheory::primesLowerThan\fR \fIN\fR
.sp
\fBmath::numtheory::primeFactors\fR \fIN\fR
.sp
\fBmath::numtheory::primesLowerThan\fR \fIN\fR
.sp
\fBmath::numtheory::primeFactors\fR \fIN\fR
.sp
\fBmath::numtheory::uniquePrimeFactors\fR \fIN\fR
.sp
\fBmath::numtheory::factors\fR \fIN\fR
.sp
\fBmath::numtheory::totient\fR \fIN\fR
.sp
\fBmath::numtheory::moebius\fR \fIN\fR
.sp
\fBmath::numtheory::legendre\fR \fIa\fR \fIp\fR
.sp
\fBmath::numtheory::jacobi\fR \fIa\fR \fIb\fR
.sp
\fBmath::numtheory::gcd\fR \fIm\fR \fIn\fR
.sp
\fBmath::numtheory::lcm\fR \fIm\fR \fIn\fR
.sp
\fBmath::numtheory::numberPrimesGauss\fR \fIN\fR
.sp
\fBmath::numtheory::numberPrimesLegendre\fR \fIN\fR
.sp
\fBmath::numtheory::numberPrimesLegendreModified\fR \fIN\fR
.sp
\fBmath::numtheory::differenceNumberPrimesLegendreModified\fR \fIlower\fR \fIupper\fR
.sp
\fBmath::numtheory::listPrimePairs\fR \fIlower\fR \fIupper\fR \fIstep\fR
.sp
\fBmath::numtheory::listPrimeProgressions\fR \fIlower\fR \fIupper\fR \fIstep\fR
.sp
.BE
.SH DESCRIPTION
.PP
This package is for collecting various number-theoretic operations, with
a slight bias to prime numbers\&.
.TP
\fBmath::numtheory::isprime\fR \fIN\fR ?\fIoption\fR \fIvalue\fR \&.\&.\&.?
The \fBisprime\fR command tests whether the integer \fIN\fR is a
prime, returning a boolean true value for prime \fIN\fR and a
boolean false value for non-prime \fIN\fR\&. The formal definition of
\'prime' used is the conventional, that the number being tested is
greater than 1 and only has trivial divisors\&.
.sp
To be precise, the return value is one of \fB0\fR (if \fIN\fR is
definitely not a prime), \fB1\fR (if \fIN\fR is definitely a
prime), and \fBon\fR (if \fIN\fR 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\&. \fB1\fR is returned for "small"
\fIN\fR (which currently means \fIN\fR < 118670087467), where it is
known that no false positives are possible\&.
.sp
The only option currently defined is:
.RS
.TP
\fB-randommr\fR \fIrepetitions\fR
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 \fIrepetitions\fR is 4\&.
.RE
.IP
Unknown options are silently ignored\&.
.TP
\fBmath::numtheory::firstNprimes\fR \fIN\fR
Return the first N primes
.RS
.TP
integer \fIN\fR (in)
Number of primes to return
.RE
.TP
\fBmath::numtheory::primesLowerThan\fR \fIN\fR
Return the prime numbers lower/equal to N
.RS
.TP
integer \fIN\fR (in)
Maximum number to consider
.RE
.TP
\fBmath::numtheory::primeFactors\fR \fIN\fR
Return a list of the prime numbers in the number N
.RS
.TP
integer \fIN\fR (in)
Number to be factorised
.RE
.TP
\fBmath::numtheory::primesLowerThan\fR \fIN\fR
Return the prime numbers lower/equal to N
.RS
.TP
integer \fIN\fR (in)
Maximum number to consider
.RE
.TP
\fBmath::numtheory::primeFactors\fR \fIN\fR
Return a list of the prime numbers in the number N
.RS
.TP
integer \fIN\fR (in)
Number to be factorised
.RE
.TP
\fBmath::numtheory::uniquePrimeFactors\fR \fIN\fR
Return a list of the \fIunique\fR prime numbers in the number N
.RS
.TP
integer \fIN\fR (in)
Number to be factorised
.RE
.TP
\fBmath::numtheory::factors\fR \fIN\fR
Return a list of all \fIunique\fR factors in the number N, including 1 and N itself
.RS
.TP
integer \fIN\fR (in)
Number to be factorised
.RE
.TP
\fBmath::numtheory::totient\fR \fIN\fR
Evaluate the Euler totient function for the number N (number of numbers
relatively prime to N)
.RS
.TP
integer \fIN\fR (in)
Number in question
.RE
.TP
\fBmath::numtheory::moebius\fR \fIN\fR
Evaluate the Moebius function for the number N
.RS
.TP
integer \fIN\fR (in)
Number in question
.RE
.TP
\fBmath::numtheory::legendre\fR \fIa\fR \fIp\fR
Evaluate the Legendre symbol (a/p)
.RS
.TP
integer \fIa\fR (in)
Upper number in the symbol
.TP
integer \fIp\fR (in)
Lower number in the symbol (must be non-zero)
.RE
.TP
\fBmath::numtheory::jacobi\fR \fIa\fR \fIb\fR
Evaluate the Jacobi symbol (a/b)
.RS
.TP
integer \fIa\fR (in)
Upper number in the symbol
.TP
integer \fIb\fR (in)
Lower number in the symbol (must be odd)
.RE
.TP
\fBmath::numtheory::gcd\fR \fIm\fR \fIn\fR
Return the greatest common divisor of \fIm\fR and \fIn\fR
.RS
.TP
integer \fIm\fR (in)
First number
.TP
integer \fIn\fR (in)
Second number
.RE
.TP
\fBmath::numtheory::lcm\fR \fIm\fR \fIn\fR
Return the lowest common multiple of \fIm\fR and \fIn\fR
.RS
.TP
integer \fIm\fR (in)
First number
.TP
integer \fIn\fR (in)
Second number
.RE
.TP
\fBmath::numtheory::numberPrimesGauss\fR \fIN\fR
Estimate the number of primes according the formula by Gauss\&.
.RS
.TP
integer \fIN\fR (in)
Number in question, should be larger than 0
.RE
.TP
\fBmath::numtheory::numberPrimesLegendre\fR \fIN\fR
Estimate the number of primes according the formula by Legendre\&.
.RS
.TP
integer \fIN\fR (in)
Number in question, should be larger than 0
.RE
.TP
\fBmath::numtheory::numberPrimesLegendreModified\fR \fIN\fR
Estimate the number of primes according the modified formula by Legendre\&.
.RS
.TP
integer \fIN\fR (in)
Number in question, should be larger than 0
.RE
.TP
\fBmath::numtheory::differenceNumberPrimesLegendreModified\fR \fIlower\fR \fIupper\fR
Estimate the number of primes between tow limits according the modified formula by Legendre\&.
.RS
.TP
integer \fIlower\fR (in)
Lower limit for the primes, should be larger than 0
.TP
integer \fIupper\fR (in)
Upper limit for the primes, should be larger than 0
.RE
.TP
\fBmath::numtheory::listPrimePairs\fR \fIlower\fR \fIupper\fR \fIstep\fR
Return a list of pairs of primes each differing by the given step\&.
.RS
.TP
integer \fIlower\fR (in)
Lower limit for the primes, should be larger than 0
.TP
integer \fIupper\fR (in)
Upper limit for the primes, should be larger than the lower limit
.TP
integer \fIstep\fR (in)
Step by which the primes should differ, defaults to 2
.RE
.TP
\fBmath::numtheory::listPrimeProgressions\fR \fIlower\fR \fIupper\fR \fIstep\fR
Return a list of lists of primes each differing by the given step from the previous one\&.
.RS
.TP
integer \fIlower\fR (in)
Lower limit for the primes, should be larger than 0
.TP
integer \fIupper\fR (in)
Upper limit for the primes, should be larger than the lower limit
.TP
integer \fIstep\fR (in)
Step by which the primes should differ, defaults to 2
.RE
.PP
.SH "BUGS, IDEAS, FEEDBACK"
This document, and the package it describes, will undoubtedly contain
bugs and other problems\&.
Please report such in the category \fImath :: numtheory\fR of the
\fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&.
Please also report any ideas for enhancements you may have for either
package and/or documentation\&.
.PP
When proposing code changes, please provide \fIunified diffs\fR,
i\&.e the output of \fBdiff -u\fR\&.
.PP
Note further that \fIattachments\fR are strongly preferred over
inlined patches\&. Attachments can be made by going to the \fBEdit\fR
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar\&.
.SH KEYWORDS
number theory, prime
.SH CATEGORY
Mathematics
.SH COPYRIGHT
.nf
Copyright (c) 2010 Lars Hellström <Lars dot Hellstrom at residenset dot net>

.fi