An Elliptic Analogue of Landau's First Problem
Abstract
Edmund Landau in 1912 questioned whether there are an infinite number of primes p of the form n^2 + 1. While this problem was and remains beyond the capabilities of modern mathematics, people have made dedicated efforts to studying distributions of these primes and elliptic analogues of similar number theoretic questions. Here, we look at the elliptic curve analogue of this problem and study the frequency that the number of solutions to an elliptic curve, over a finite field with prime order, is a perfect square. We examine some of the anomalies in the distribution of square solutions, investigate whether this distribution is similar to that conjectured of squares of the form p − 1, and translate previous conjectures on the frequencies of fixed N_p values to our square problem.