Triangles with prime hypotenuse

TY - JOUR

T1 - Triangles with prime hypotenuse

AU - Chow, Samuel Khai Ho

AU - Pomerance, Carl

N1 - ©The Author(s) 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

PY - 2017/10/9

Y1 - 2017/10/9

N2 - 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 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.

AB - 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 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.

KW - Gaussian primes

KW - Pythagorean triples

U2 - 10.1007/s40993-017-0086-6

DO - 10.1007/s40993-017-0086-6

M3 - Article

VL - 3

JO - Research in Number Theory

JF - Research in Number Theory

M1 - 21

ER -