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A sequence of primes q_1<q_2<...<q_k is a Cunningham chain of the first kind (second kind) of length k if q_(i+1)=2q_i+1 (q_(i+1)=2q_i-1) for i=1, ..., k-1. Cunningham primes ...

Number Theory

There are three types of cubic lattices corresponding to three types of cubic close packing, as summarized in the following table. Now that the Kepler conjecture has been ...

The function frac(x) giving the fractional (noninteger) part of a real number x. The symbol {x} is sometimes used instead of frac(x) (Graham et al. 1994, p. 70; Havil 2003, ...

For a given m, determine a complete list of fundamental binary quadratic form discriminants -d such that the class number is given by h(-d)=m. Heegner (1952) gave a solution ...

There exists a positive integer s such that every sufficiently large integer is the sum of at most s primes. It follows that there exists a positive integer s_0>=s such that ...

A tetradic (or four-way) number is a number that remains unchanged when flipped back to front, mirrored up-down, or flipped up-down. Since the only numbers that remain ...

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

Grimm conjectured that if n+1, n+2, ..., n+k are all composite numbers, then there are distinct primes p_(i_j) such that p_(i_j)|(n+j) for 1<=j<=k.

The Schnirelmann density of a set of nonnegative integers is the greatest lower bound of the fractions A(n)/n where A(n) is the number of terms in the set <=n.