Number Theory: Prime Numbers - Maple Programming Help

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Number Theory: Prime Numbers

Getting Started

While any command in the package can be referred to using the long form, for example, NumberTheory:-Divisors, it is often easier to load the package and then use the short form command names.

restart;

with(NumberTheory):

Examples

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The top-level ithprime command returns the ith prime. For example, the first ten primes are given by the following sequence:

seqithprimei,i=1..10

2,3,5,7,11,13,17,19,23,29

(1)

The top-level isprime command determines if a given number is prime:

isprime28

false

(2)

isprime29

true

(3)

The Divisors command can verify that a number is prime. If the divisors of a given integer are only 1 and itself, the number is prime.

Divisors28

1,2,4,7,14,28

(4)

Divisors29

1,29

(5)

The SumOfDivisors command returns the sum of the divisors of an integer:

SumOfDivisors28

56

(6)

The top-level ifactor command gives the integer factorization of an integer:

ifactor28

227

(7)

The PrimeFactors command returns a list of factors for a given integer that are primes without multiplicity:

PrimeFactors28

2,7

(8)

The NumberOfPrimeFactors command returns the number of prime factors of an integer, counted with multiplicity:

NumberOfPrimeFactors28

3

(9)

The Radical command returns the product of the prime divisors of an integer:

Radical28

14

(10)

The top-level nextprime (or prevprime) command returns the next (or previous) prime numbers after (or before) the given integer:

nextprime29

31

(11)

The PrimeCounting command returns the number of primes less than a given integer:

PrimeCounting31

11

(12)

Two integers are relatively prime (coprime) if the greatest common divisor of the values is 1. The AreCoprime command tests if a sequence of integers or Gaussian integers are coprime:

AreCoprime5,8

true

(13)

AreCoprime2,8

false

(14)

The following plot shows the coprimes for the integers 1 to 15:

Statistics:-HeatMapMatrix15,i,j`if`AreCoprimei,j,1,0,color=White,Red

An integer is called square-free if it is not divisible by the square of another number other than 1. All prime numbers are square-free:

IsSquareFree31

true

(15)

IsSquareFree7101

true

(16)

IsSquareFree372

false

(17)

See Also

NumberTheory