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The first Hardy-Littlewood conjecture is called the k-tuple conjecture. It states that the asymptotic number of prime constellations can be computed explicitly. A particular ...

A random number generator produced by iterating X_(n+1)=|100lnX_n (mod 1)| for a seed X_0=0.1. This simple generator passes the noise sphere test for randomness by showing no ...

A prime factorization algorithm which can be implemented in a single-step or double-step form. In the single-step version, a prime factor p of a number n can be found if p-1 ...

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 ...

Each prime factor p_i^(alpha_i) in an integer's prime factorization is called a primary.

If f_1(x), ..., f_s(x) are irreducible polynomials with integer coefficients such that no integer n>1 divides f_1(x), ..., f_s(x) for all integers x, then there should exist ...

The prime number theorem shows that the nth prime number p_n has the asymptotic value p_n∼nlnn (1) as n->infty (Havil 2003, p. 182). Rosser's theorem makes this a rigorous ...

Lehmer's totient problem asks if there exist any composite numbers n such that phi(n)|(n-1), where phi(n) is the totient function? No such numbers are known. However, any ...

Hoffman (1998, p. 90) calls the sum of the exponents in the prime factorization of a number its roundness. The first few values for n=1, 2, ... are 0, 1, 1, 2, 1, 2, 1, 3, 2, ...

In an integral domain R, the decomposition of a nonzero noninvertible element a as a product of prime (or irreducible) factors a=p_1...p_n, (1) is unique if every other ...