On the Diophantine Equation 29^x-31^y=z^3.

Apisit Pakapongpun, Natdanai Chailangka

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Published: Jan 1, 2026

DOI: 10.69793/ijmcs/04.2026/chailangka

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Source abstract

In this paper, we study the exponential Diophantine equation 29^x-31^y=z^3, where x,y,z are non-negative integers. We show that 6\mid x by using congruences modulo 8, 7, 9 and 13. Then we use a quadratic congruence modulo 31 to show that (x,y,z)=(0,0,0) is the only solution.

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