# Triangles with prime hypotenuse

## TL;DR

This paper investigates the distribution of odd and even legs in right triangles with integer sides and prime hypotenuse, establishing that their density tends to zero and providing bounds using advanced number theory techniques.

## Contribution

It introduces new bounds on the density of such triangles' legs, combining sieve methods, Gaussian prime distribution, and Hardy--Ramanujan inequality, advancing understanding of their rarity.

## Key findings

- Upper density of such triangles is zero with logarithmic decay.
- Provides a nontrivial lower bound for the sequence.
- Uses advanced sieve and prime distribution techniques.

## Abstract

The sequence $3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\dots$ consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erd\H{o}s--Ford--Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes in narrow sectors, and the Hardy--Ramanujan inequality.