Intersections of binary quadratic forms in primes and the paucity phenomenon

Rational Representations of Primes by Binary Quadratic Forms

Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u2 + qv2 has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that pk2 ∈ N , where N is the set of norms of algebraic integers in Q( √ −q). Let k(q) denote the smallest positive integer k such that pk2 ∈ N for all q-admissible primes p. It is shown that ...

Binary Quadratic Forms , Genus Theory , and Primes of the Form p = x 2 + ny 2

In this paper, we will develop the theory of binary quadratic forms and elementary genus theory, which together give an interesting and surprisingly powerful elementary technique in algebraic number theory. This is all motivated by a problem in number theory that dates back at least to Fermat: for a given positive integer n, characterizing all the primes which can be written p = x + ny for some...

Division and Binary Quadratic Forms

has only three elements, written h(−23) = 3. There is an binary operation called composition that takes two primitive forms of the same discriminant to a third. Composition is commutative and associative, and makes the set of forms into a group, with identity 〈1, 0,−∆/4〉 for even discriminant and 〈1, 1, (1−∆)/4〉 for odd. From page 49 of Buell [1]: if a form 〈α, β, γ〉 represents a number r primi...

Composition of Binary Quadratic Forms

Positive Definite Binary Quadratic Forms That Represent Almost the Same Primes

Jagy and Kaplansky exhibited a table of 68 pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that this list is complete outside of “trivial” pairs. In this article, we find all pairs of such forms that represent the same primes outside of a finite set.