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

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 ($A=5^{\alpha}, ~B=64,~ c=3, ~m=2; \quad A=2^{\alpha},~ B=27, ~c \in \{7,13\}, ~m=3$).

Publication Details

Journal
ArXiv.org
Published
2025-04-20
Primary Topic
Advanced Mathematical Theories and Applications
Type
article
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article

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

ArXiv.org
Advanced Mathematical Theories and Applications
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On some ternary Diophantine equations of Signature $(p,p,k)$

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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 ($A=5^{\alpha}, ~B=64,~ c=3, ~m=2; \quad A=2^{\alpha},~ B=27, ~c \in \{7,13\}, ~m=3$).

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Advanced Mathematical Theories and Applications
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On some ternary Diophantine equations of Signature $(p,p,k)$ · ArXiv.org (2025) | TGRS Research Map | TGRS