A binary quadratic approach to $X^2+(2k-1)^Y=k^Z$

TL;DR

This paper proves Terai's conjecture for specific cases using class group structures of binary quadratic forms, confirming the unique solution for certain values of k where 4 divides k and 2k-1 is a prime power.

Contribution

It introduces a novel approach leveraging class group structures to prove the conjecture for cases with 4 dividing k and 2k-1 being a prime power.

Findings

Proves Terai's conjecture for 4 dividing k and 2k-1 prime power

Establishes the conjecture for 4 ≤ k ≤ 1000

Demonstrates the effectiveness of class group methods in exponential Diophantine equations

Abstract

A conjecture of N. Terai states that for any integer $k>1$, the equation $x^2+(2k-1)^y =k^z$ has only one solution, namely, $(x, y, z) = (k-1, 1, 2).$ Using the structure of class groups of binary quadratic forms, we prove the conjecture when $4\Vert k$, with $2k-1$ a prime power and $4\le k\le 1000$.