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Certain finite simple groups can be recognized by the degrees of the vertices in their prime graphs. [3] The connected components of a prime graph have diameter at most five, and at most three for solvable groups. [4] When a prime graph is a tree, it has at most eight vertices, and at most four for solvable groups. [5]
If an AP-k does not begin with the prime k, then the common difference is a multiple of the primorial k# = 2·3·5·...·j, where j is the largest prime ≤ k. Proof: Let the AP-k be a·n + b for k consecutive values of n. If a prime p does not divide a, then modular arithmetic says that p will divide every p'th term of the arithmetic ...
The only base-4 repunit prime is 5 (). = (+) (), and 3 always divides + when n is odd and when n is even. For n greater than 2, both + and are greater than 3, so removing the factor of 3 still leaves two factors greater than 1.
This category is for articles about classes (meaning subsets here) of prime numbers, for example primes generated by a particular formula or having a special property. See List of prime numbers for definitions and examples of many classes of primes.
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
A prime number is a natural number that has no natural number divisors other than the number 1 and itself.. To find all the prime numbers less than or equal to a given integer N, a sieve algorithm examines a set of candidates in the range 2, 3, …, N, and eliminates those that are not prime, leaving the primes at the end.
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In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d.