Number Theory: Prime Numbers
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.
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:
The top-level isprime command determines if a given number is prime:
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.
The SumOfDivisors command returns the sum of the divisors of an integer:
The top-level ifactor command gives the integer factorization of an integer:
The PrimeFactors command returns a list of factors for a given integer that are primes without multiplicity:
The NumberOfPrimeFactors command returns the number of prime factors of an integer, counted with multiplicity:
The Radical command returns the product of the prime divisors of an integer:
The top-level nextprime (or prevprime) command returns the next (or previous) prime numbers after (or before) the given integer:
The PrimeCounting command returns the number of primes less than a given integer:
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:
The following plot shows the coprimes for the integers 1 to 15:
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:
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