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

Maohua Le; Anitha Srinivasan

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

Number Theory 2023-12-05 v2

Authors: Maohua Le , Anitha Srinivasan

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 $4\le k\le 1000$ .

Keywords

exponential diophantine equation terai conjecture binary quadratic forms class groups prime power

Cite

@article{arxiv.2210.04758,
  title  = {A binary quadratic approach to $X^2+(2k-1)^Y=k^Z$},
  author = {Maohua Le and Anitha Srinivasan},
  journal= {arXiv preprint arXiv:2210.04758},
  year   = {2023}
}

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