On some ternary Diophantine equations of Signature $(p,p,k)$

TL;DR

The paper advances the modular-method framework for ternary Diophantine equations with signature $(p,p,m)$ by treating three explicit forms $5^{\alpha}x^n+64y^n=3z^2$, $2^{\alpha}x^n+27y^n=7z^3$, and $2^{\alpha}x^n+27y^n=13z^3$ with prime $n$ and suitable $m$, establishing nonexistence of nonzero coprime solutions under stated conditions. It constructs Frey elliptic curves associated to putative solutions, applies level-lowering to weights-2 newforms at computed levels, and uses Norm congruence criteria to eliminate candidate newforms, complemented by computational Mordell-curve checks in exceptional cases. The results extend the reach of the BS and BVY methodologies to specific coefficient patterns and demonstrate how a combination of modular techniques and practical arithmetic (Magma computations) yields explicit insolubility criteria for these exponential equations. Overall, the work contributes concrete insolubility results for particular ternary equations and illustrates a robust protocol for handling similar problems via modularity, conductor analysis, and explicit coefficient tests.

Abstract

In this paper, we summarize the work on ternary Diophantine equation of the form $Ax^n+By^n=cz^m$, where $m \in \{2,3,n\} $, $n\geqslant 7 $ is a prime. Moreover, we completely solve some particular cases ($A=5^α, ~B=64,~ c=3, ~m=2; \quad A=2^α,~ B=27, ~c \in \{7,13\}, ~m=3$).

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Proposition 9
• Proposition 10