Triangles with prime hypotenuse

Abstract

The sequence 3, 5, 9, 11, 15, 19, 21, 25, 29, 35, . . . 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 Erdős-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.

Symmetric primes revisited

“…(In fact, Ford [5] has further shown that if the implied constant O(1) is replaced with −3/2, the resulting expression has the same magnitude as M (N ).) A more recent appearance of η occurs in Chow and Pomerance [3], where the odd legs in integer-sided right triangles with prime hypotenuse are considered. (The present note uses some techniques from [3].…”

“…A more recent appearance of η occurs in Chow and Pomerance [3], where the odd legs in integer-sided right triangles with prime hypotenuse are considered. (The present note uses some techniques from [3]. )…”