A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x^{2}+b^{y}=c^{z}

TL;DR

This paper proves Terai's conjecture for the exponential Diophantine equation x^{2}+b^{y}=c^{z} when b is a product of two primes and c is congruent to 5 modulo 8, confirming the conjecture in this specific case.

Contribution

The paper establishes the validity of Terai's conjecture for a new class of cases where b is a product of two primes and c ≡ 5 (mod 8).

Findings

Terai's conjecture holds for b as a product of two primes.

Confirmed the conjecture for c ≡ 5 (mod 8).

Extended understanding of solutions to the exponential Diophantine equation.

Abstract

Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine equation x^{2}+b^{y}=c^{z} has only the positive integer solutions (x,y,z)=(a,2,2). In this study, we prove that Terai's conjecture is true when b is a product of two primes and c=5(mod8).