Counting prime numbers in arithmetic sequences

Prime Numbers in Certain Arithmetic Progressions

We discuss to what extent Euclid’s elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet’s theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod k ) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.

Prime numbers with Beatty sequences

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form p = 2 ⌊αn⌋+1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a ...

Gaps between Prime Numbers and Primes in Arithmetic Progressions

The equivalence of the two formulations is clear by the pigeon-hole principle. The first one is psychologically more spectacular: it emphasizes the fact that for the first time in history, one has proved an unconditional existence result for infinitely many primes p and q constrained by a binary condition q − p = h. Remarkably, this already extraordinary result was improved in spectacular fashi...

Counting Prime Numbers with Short Binary Signed Representation

Modular arithmetic with prime moduli has been crucial in present day cryptography. The primes of Mersenne, Solinas, Crandall and the so called IKE-MODP have been extensively used in efficient implementations. In this paper we study the density of primes with binary signed representation involving a small number of non-zero ±1-digits.

Construction of normal numbers via pseudo-polynomial prime sequences