Overview of the NumberTheory Package - Maple Programming Help

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Overview of the NumberTheory Package

 Calling Sequence NumberTheory[command](arguments) command(arguments)

Description

 • The NumberTheory package contains commands used to investigate the properties of the natural numbers and integers (see number theory).
 • Each command in the NumberTheory package can be accessed by using either the long form or the short form of the command name in the calling sequence.
 As the underlying implementation of the NumberTheory package is a module, it is also possible to use the form NumberTheory:-command to access a command from the package. For more information, see Module Members.

List of NumberTheory Package Commands

 • The following is a list of commands in the NumberTheory package.

 test whether a sequence of numbers is relatively prime compute the nth term in the Calkin-Wilf sequence Carmichael's lambda function continued fraction expansion simple continued fraction expansions for real roots of a rational polynomial minimal polynomials of primitive roots of unity with rational coefficients the set of positive divisors of an integer factorization of integers in quadratic norm-Euclidean fields solution to Minkowski's linear forms modular square root of -1 inhomogeneous Diophantine approximation integral base of an algebraic number field inverse of Euler's totient function test whether a polynomial is cyclotomic test whether a number is a Mersenne number test whether an integer is square free Mersenne exponents generalized Legendre symbol generalized Jacobi symbol largest integer power divisor of a number quadratic residuosity solutions to the modulo n extended GCD problem discrete logarithm under modular arithmetics modular root modular square root Möbius function order of a number under modular multiplication solution to the nearby lattice point problem least safe prime greater than a number number of monic irreducible polynomial number of prime factors counted with multiplicity number of prime numbers less than a number prime factors of an integer primitive root modulo n pseudo primitive root modulo n quadratic residuosity of a number rational number in repeating decimal form modular roots of unity sum of powers of the divisors solutions to the sum of two squares problem solutions to a Thue equation or inequality Euler's totient function

Examples

 > $\mathrm{with}\left(\mathrm{NumberTheory}\right):$

Show $\frac{3}{7}$ as a repeated decimal:

 > $\mathrm{RepeatingDecimal}\left(\frac{3}{7}\right)$
 ${\mathrm{0.\left(428571\right)}}$ (1)

Show $\frac{3}{7}$ as a continued fraction:

 > $\mathrm{ContinuedFraction}\left(\frac{3}{7}\right)$
 ${0}{+}\frac{{1}}{{2}{+}\frac{{1}}{{3}{+}{0}}}$ (2)

Euler's totient (phi) function is an arithmetic function that counts the positive integers less than or equal to a given value, $n$, that are coprime to $n$. The PrimeCounting (or pi) command returns the number of primes less than an integer, $n$.

Comparing pi(n) with phi(n) for the first forty values for n:

 > $\mathrm{plots}:-\mathrm{display}\left(\mathrm{DynamicSystems}:-\mathrm{DiscretePlot}\left(\left[\mathrm{seq}\left(i-0.1,i=2..40\right)\right],\left[\mathrm{seq}\left(\mathrm{PrimeCounting}\left(n\right),n=2..40\right)\right],\mathrm{style}=\mathrm{stem},\mathrm{symbol}=\mathrm{soliddiamond},\mathrm{color}="Crimson",\mathrm{legend}="pi"\right),\mathrm{DynamicSystems}:-\mathrm{DiscretePlot}\left(\left[\mathrm{seq}\left(i+0.1,i=2..40\right)\right],\left[\mathrm{seq}\left(\mathrm{Totient}\left(n\right),n=2..40\right)\right],\mathrm{style}=\mathrm{stem},\mathrm{symbol}=\mathrm{solidcircle},\mathrm{color}="MidnightBlue",\mathrm{legend}="Totient"\right),\mathrm{linestyle}=\mathrm{dot},\mathrm{transparency}=0.1,\mathrm{size}=\left[800,400\right],{\mathrm{axis}}_{2}=\left[\mathrm{gridlines}=\left[\mathrm{thickness}=0,\mathrm{color}="LightGrey"\right]\right]\right)$

Two integers are relatively prime (coprime) if the greatest common divisor of the values is 1. The following plot shows the coprimes for the integers 1 to 25:

 > $\mathrm{Statistics}:-\mathrm{HeatMap}\left(\mathrm{Matrix}\left(25,\left(i,j\right)→\mathrm{if}\left(\mathrm{AreCoprime}\left(i,j\right),1,0\right)\right),\mathrm{color}=\left["White","Pink"\right]\right)$

Compatibility

 • The NumberTheory package was introduced in Maple 2016.