A parametrization of $\theta$-congruent numbers with many prime factors and with prescribed prime factors

Abstract

Let theta be a real number such that 0 < theta < pi and cos theta is an element of Q. For each positive integer n, we give a parametrization S-n (alpha) whose square-free part N-n (alpha) for each negative integer alpha is a theta-congruent number with many prime factors including any given primes (especially, at least n prime factors that are guaranteed to appear) by showing the positivity of the rank of the corresponding theta-congruent number elliptic curve over Q. Especially, we show that if a given odd prime p > 2n is near 2n, then p appears as a factor of N-n(alpha) very often as a varies all over negative integers by proving that the probability of the set of all negative integers alpha such that p divides N-n (alpha) is 2n+1/p+1. (C) 2018 Elsevier Inc. All rights reserved.