On the Diophantine equation (pⁿ)ˣ + (4ᵐ + p)ʸ = z² when p, 4ᵐ + p are prime integers

Nam, Pham Hong (2024) On the Diophantine equation (pⁿ)ˣ + (4ᵐ + p)ʸ = z² when p, 4ᵐ + p are prime integers. ANNALES MATHEMATICAE ET INFORMATICAE, 60. pp. 108-120. ISSN 1787-6117

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Abstract

In this paper, we give methods to solve the Diophantine equation (pn)x + (p + 4m)y = z2 where p ≥ 3 and p + 4m are prime integers. Concretely, using the congruent method, one proves that this equation has no non-negative solutions if p > 3. For the case p = 3, using the elliptic curves, we will show that this equation has no solutions if m ≥ 3. In this case, when m = 1 using the elliptic curves, we will show that this equation has only solution (x,y,z) = (2,1,4) if n = 1 and (x,y,z) = (1,1,4) if n = 2, and when m = 2 using the elliptic curves, we will show that this equation has only solutions is (x,y,z) = (4,1,10) if n = 1 and (x,y,z) = (2,1,10) if n =2.

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