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One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to ...
Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. [1] It is constructed by writing the positive integers in a square spiral and specially marking the prime ...
A prime p is called a Wolstenholme prime iff the following condition holds: ().If p is a Wolstenholme prime, then Glaisher's theorem holds modulo p 4.The only known Wolstenholme primes so far are 16843 and 2124679 (sequence A088164 in the OEIS); any other Wolstenholme prime must be greater than 10 11. [2]
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers. A composite number has Ω(n) > 1.
Fermat's little theorem states that if p is prime and a is not divisible by p, then (). If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds. If it does not hold for a value of a, then p is composite.
A theorem states that n is prime if and only if all such functions p n are algebra endomorphisms. In-between these two conditions lies the definition of Carmichael number of order m for any positive integer m as any composite number n such that p n is an endomorphism on every Z n-algebra that can be generated as Z n-module by m elements ...
See also Theor.2.3 in ``Regularities of Twin, Triplet and Multiplet Prime Numbers," arXiv:1103.0447[math.NT], Global J.P.A.Math 8(2012), in press.) If the AP is prime for k consecutive values, then a must therefore be divisible by all primes p ≤ k. This also shows that an AP with common difference a cannot contain more consecutive prime terms ...