This package is for collecting various number-theoretic operations, with a slight bias to prime numbers.
math::numtheory::isprimeN ?optionvalue ...?
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:
-randommrrepetitions
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::firstNprimesN
Return the first N primes
integer N (in)
Number of primes to return
math::numtheory::primesLowerThanN
Return the prime numbers lower/equal to N
integer N (in)
Maximum number to consider
math::numtheory::primeFactorsN
Return a list of the prime numbers in the number N
integer N (in)
Number to be factorised
math::numtheory::primesLowerThanN
Return the prime numbers lower/equal to N
integer N (in)
Maximum number to consider
math::numtheory::primeFactorsN
Return a list of the prime numbers in the number N
integer N (in)
Number to be factorised
math::numtheory::uniquePrimeFactorsN
Return a list of the unique prime numbers in the number N
integer N (in)
Number to be factorised
math::numtheory::factorsN
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::totientN
Evaluate the Euler totient function for the number N (number of numbers relatively prime to N)
integer N (in)
Number in question
math::numtheory::moebiusN
Evaluate the Moebius function for the number N
integer N (in)
Number in question
math::numtheory::legendreap
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::jacobiab
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::gcdmn
Return the greatest common divisor of m and n
integer m (in)
First number
integer n (in)
Second number
math::numtheory::lcmmn
Return the lowest common multiple of m and n
integer m (in)
First number
integer n (in)
Second number
math::numtheory::numberPrimesGaussN
Estimate the number of primes according the formula by Gauss.
integer N (in)
Number in question, should be larger than 0
math::numtheory::numberPrimesLegendreN
Estimate the number of primes according the formula by Legendre.
integer N (in)
Number in question, should be larger than 0
math::numtheory::numberPrimesLegendreModifiedN
Estimate the number of primes according the modified formula by Legendre.
integer N (in)
Number in question, should be larger than 0
math::numtheory::differenceNumberPrimesLegendreModifiedlowerupper
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::listPrimePairslowerupperstep
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::listPrimeProgressionslowerupperstep
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
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