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The series producing Brun's constant converges even if there are an infinite number of twin primes, first proved by Brun (1919).

The Mertens constant B_1, also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant (Bach and Shallit 1996, p. 234), or Kronecker's constant ...

A number n is called an economical number if the number of digits in the prime factorization of n (including powers) uses fewer digits than the number of digits in n. The ...

A bitwin chain of length one consists of two pairs of twin primes with the property that they are related by being of the form: (n-1,n+1) and (2n-1,2n+1). (1) The first few ...

Every even number is the difference of two consecutive primes in infinitely many ways (Dickson 2005, p. 424). If true, taking the difference 2, this conjecture implies that ...

The first of the Hardy-Littlewood conjectures. The k-tuple conjecture states that the asymptotic number of prime constellations can be computed explicitly. In particular, ...

A prime constellation, also called a prime k-tuple, prime k-tuplet, or prime cluster, is a sequence of k consecutive numbers such that the difference between the first and ...

A prime gap of length n is a run of n-1 consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes p_k and p_(k+1) ...

A prime p is said to be a Sophie Germain prime if both p and 2p+1 are prime. The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ... (OEIS ...

Every "large" even number may be written as 2n=p+m where p is a prime and m in P union P_2 is the set of primes P and semiprimes P_2.