A Theorem of Congruent Primes

TL;DR

This paper introduces a new theorem to determine whether a prime number is a congruent number, using a binary tree method to analyze solutions, which may have broader applications beyond primes.

Contribution

The paper presents a novel proof method involving binary trees to analyze congruent primes, offering potential for generalization to non-prime numbers.

Findings

Theorem provides a new criterion for congruent primes.

The proof method tracks solution sets as binary trees.

Contradiction arises for primes, indicating a new approach to the problem.

Abstract

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter than existing proofs, but the method may be useful in other contexts. The proof of Theorem 1 tracks the set of solutions and this set branches as a binary tree. Conditions set to the theorem restricts the branches so that only one branch is left. Following this branch gives either a solution or a contradiction. In Theorem 1 it leads to a contradiction. The interest is in the proof method, which maybe can be generalized to non-primes.