Changes in version 0.9-2 (2025-11-20) o polygonal() generates sequences of polygonal numbers, including triangular, square, pentagonal and hexagonal numbers. o arithmetic_progression() generates arithmetic progressions. o unitary divisors are introduced on the help page of divisors(). Changes in version 0.8-9 o modNthroot() finds n-th roots of integers modulo a prime number. Changes in version 0.8-7 o cf2num() improved, scaled version, replacing the alternating series. o num2cf() direct implementation that allows for 'mpfr' numbers. Changes in version 0.8-5 (2022-11-23) o periodicCF() determines the the full and exact period of the periodic continued fraction of sqrt(d), d a non-square integer. o solvePellsEq() returns the fundamental solution of Pell's equation. Changes in version 0.8-4 o contFrac() renamed to contfrac(). o contfrac() additionally returns the convergents 'p' and 'q'. o chinese() and isNatural(): Corrected "coerce to logical" bug. Changes in version 0.8-3 o Added isSquare() and isSquarefree() for convenience. o Changed the default tolerance of contFrac() to 1e-12. o Corrected some misuses of '&' resp '|'. Changes in version 0.8-2 (2021-05-14) o Removed example for farey_seq() from the 'ratFarey' help page because apparently 'gmp' causes problems for macOS on CRAN servers. Changes in version 0.8-1 (2021-04-11) o carmichael() determines Carmichael numbers. o Corrected all old and 'insecure' Internet links with 'https:'. Changes in version 0.7-9 o stern_brocot_seq() generates the Stern-Brocot sequence (fast). o farey_seq() generates the full n-th Farey series (slow). Changes in version 0.7-8 o bernoulli_numbers() the Bernoulli numbers w/ and w/o 'big rationals'. o pascal_triangle() generates Pascal numbers in a rectangle. o modq() extends the modulo operator to rational numbers. Changes in version 0.7-6 o Help clarified for omega() and Omega() functions. Changes in version 0.7-5 (2019-11-26) o collatz() calculates generalized Collatz sequences. Changes in version 0.7-4 o "length > 1 in coercion to logical" error corrected in mGCD() and mLCM(). o Corrected hermiteNF(), suggested by Martin Hoffmann. Changes in version 0.7-3 o modsqrt() calculates the square root modulo primes. Changes in version 0.7-1 (2018-05-17) o Removed 'numbers-package.Rd' on request of K. Hornick, CRAN. Changes in version 0.6-8 o intnthroot() calculates the integer n-th root. Changes in version 0.6-7 o modlog() the modular (or: discrete) logarithm. o primroot() got a new keyword 'all=FALSE' to return all primitive roots if it is TRUE. Also, isPrimroot() with the obvious meaning. Changes in version 0.6-6 (2017-01-10) o Extended the description line considerably by request of CRAN. o Finally completed the "?`numbers-package`" entry of the help. Changes in version 0.6-5 o cf2num() converts (generalized) continued fractions to numbers, with special care for approximating infinite fractions. Changes in version 0.6-3 o divisors() lists all divisors of a number n from its prime factors. o necklace() and bracelet() compute the number of necklaces resp. bracelets in combinatorics, suggested by David Sterratt. o corrected a 'tiny' bug in modpower(), pointed out by Nathan Carter. Changes in version 0.6-1 (2015-07-14) o bell() generates Bell numbers. o Spelling changes in the documentation. Changes in version 0.5-9 (2015-07-10) o Changed package 'gmp' status from "Imports:" to "Suggests:"; functions miller_rabin() and mersenne() require 'gmp' to be loaded. o sigma() renamed to Sigma() to avoid name clash. Changes in version 0.5-8 o atkin_sieve(): Atkin's prime number sieve. o Small bug corrected: eulerPhi(1) == 1 . Changes in version 0.5-6 (2015-03-14) o Pi-day 3.14.15 9:26:53.58 contribution: dropletPi() realizes the droplet/spigot algorithm for pi; droplet_e() has been renamed to dropletE(). Changes in version 0.5-3 o radical() computes the radical of n, i.e the product of unique prime factors of n. Changes in version 0.5-2 (2015-01-28) o miller-rabin() executes the probabilistic Miller-Rabin primality test, faster than isPrime(), but still slower than gmp::isprime(). Changes in version 0.5-1 o egyptian_complete() returns the number of solutions found. o legendre_sym() returned Boolean nonsense, has been corrected. Changes in version 0.4-9 o ordpn() order of a prime number in n!, i.e. n faculty. o fibonacci() and lucas() corrected; the recursive computation has been replaced by an iterative approach. Changes in version 0.4-7 o agm() exact to machine accuracy; returns only the AGM value. Changes in version 0.4-5 (2014-01-08) o Imports 'gmp'. o Primes() avoids creating additional memory, doubled its speed. Changes in version 0.4-3 o legendre_sym() Legendre and Jacobi symbol. o quadratic_residues() lists all quadratic residues. Changes in version 0.4-1 o mersenne() computes Mersenne prime numbers. o Renamed factorize() to primeFactors() (avoid masking ...) Changes in version 0.3-5 o catalan() Catalan numbers. o pythagorean_triple() generating Pythagorean triples. Changes in version 0.3-3 (2012-11-21) o hermiteNF() Hermite normal form. o lucas() Lucas numbers as sequence. o Added corrections to mGCD() and mLCM(). Changes in version 0.3-1 (2012-10-05) o chinese() Chinese Remainder Theorem. o egypt_methods(), egypt_complete() Egyptian fractions o zeck() Zeckendorf representation. o Improving modular arithmetics: mod(), rem(), div(). Changes in version 0.2-1 o agm() algebraic-geometric mean. o fibonacci() Fibonacci sequence. o droplet_e() for generating digits of e. o Modular functions: - modinv(), modlin() modular inverses; - primroot() primitive roots. o Greatest common divisor, least common multiple: - extGCD(), GCD(), mGCD(), LCM(), mLCM(), coprime(). Changes in version 0.1-1 o More Number-theoretic functions: - eulersPhi; moebius(), mertens(); - sigma(), tau(), omega(), Omega(). o Shifted number-theoretic functions from 'pracma' to 'numbers': - contFrac() continuous fractions; - ratFarey() rational approximation through Farey sequence. o Prime number functions: - primeSieve(), Primes(), isPrime(), factorize(); - twinPrimes(), nextPrime(), previousPrime(); - isNatural(), isIntpower(). o New package 'numbers' on R-Forge.