The Number of Representations Function for Positive Binary Quadratic Forms

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n= x2 +Ny2 for a squarefree integer N .

Lectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms

Notice that this means if we change variables ( x′ y′ ) = v′ = Av (the entries of A are integers) then Q(x′, y′) = (Av) QAv = v (A QA)v. Therefore, if Q represents a number then so does A QA. In particular, if A has an inverse with integer entries, then we get that Q and A QA represent all the same numbers. Clearly if A has an inverse B with integer entries, then det A detB = det AB = 1, thus d...

On the Number of Perfect Binary Quadratic Forms

REPRESENTATIONS OF DEFINITE BINARY QUADRATIC FORMS OVER F q [ t ]

In this paper, we prove that a binary definite quadratic form over Fq[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m− 2, where m is the degree of its discriminant. We also characterize, when q > 13, all the definite binary forms over Fq[t] that have class number one.

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 ...