The distribution of prime ideals of imaginary quadratic fields

Abstract

Abstract. Let Q(x, y) be a primitive positive definite quadratic form with integer coefficients. Then, for all (s, t) ∈ R 2 there exist (m, n) ∈ Z 2 such that Q(m, n) is prime andThis is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.

Infinite Primes, Quadratic Polynomials, and Fermat’s Criterion

“…In fact, there is a very great probability for appearing of primes of the form 2x 2 -1, because there are 44 x-values less 100 i.e. x=2, 3,4,6,7,8,10,11,13,15,17,18,21,22,24,25,28,34,36,38,39,41,42,45,46,49,50,52,56,59,62,63,64,69,73,76,80,81,85,87,91,92,95,98 to generate primes of the form 2x 2 -1. More such primes can be viewed in the on-line Encyclopedia of Integer Sequences.…”

Diophantine approximation with arithmetic functions, I

“…A sketch of the proof of Theorem 3 will appear in section 4, together with a discussion of how to deal with more general f i and g i . Likely the methods in [3] can be used to prove Γ ′ ≤ 0.525, but the best result available in the literature is Γ ′ ≤ 0.53 (by considering the polynomial Q(x, y) = x 2 + y 2 in [13]).…”

Gaussian primes in almost all narrow sectors

“…The problem of Gaussian primes in small balls is similar in flavour to that of primes in short intervals, and was studied by Coleman [1] who established individual results, both on GRH and unconditionally (see also [4]), and a result about almost all balls.…”