On the exponential Diophantine equation p⋅ 3^x+p^y=z² with p a prime number

2023/08/19 by Anderson L. P. Porto, Porto, A. L. P., Marcelo Buosi +3
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2308.10043

openalex publication_date 2023/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we find non-negative integer solutions for exponential Diophantine equations of the type p ⋅ 3^x+ p^y=z², where p is a prime number. We prove that such equation has a unique solution (x,y,z)=(log₃(p-2), 0, p-1) if 2 ≠ p ≡ 2 \pmod 3 and (x,y,z)=(0,1,2) if p=2. We also display the infinite solution set of that equation in the case p=3. Finally, a brief discussion of the case p ≡ 1 \pmod 3 is made, where we display an equation that does not have a non-negative integer solution and leave some open questions. The proofs are based on the use of the properties of the modular arithmetic.

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