On the Equations 2m2 +2m = yn and M(m+2) =yn: Connections to Quadratic Forms and Major Conjectures in Number Theory

Mar Detic

On the Equations 2m2 +2m = yn and M(m+2) =yn: Connections to Quadratic Forms and Major Conjectures in Number Theory

This paper provides a comprehensive analysis of the Diophantine equations 2m2+2m =yn andm(m+2) =yn forintegersm,y ≥ 0andn ≥ 2. Wedemonstrate that the first equation has infinitely many solutions for n = 2 (via a Pell equation) and only the trivial solution for n ≥ 3 (by Erd˝os—Selfridge), while the second has no nontrivial solutions for any n ≥ 2. We explore connections to Fermat’s Last Theorem, the Beal Conjecture, and the ABC Conjecture. Additionally, we show that for odd m = 2k+1, the equation m(m+2) = yn becomes 4(k+1)2−1 = yn, connecting it to arithmetic progressions and Pell-type equations. We demonstrate that attempts to express these equations in Beal form fail, and we highlight the role of discriminants and factorization in determining the existence of solutions

Comments: 7 Pages.

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[v1] 2025-09-09 12:19:39

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