A Diophantine Identity of Sophie Germain Type and a Telescoping Family of Series with Zeta Asymptotics

Resumo

We study a Diophantine equation encoding a family of factorizations of Sophie Germain type for binomials of the form \[ (a_1 x^n)^2 + (a_2 y^n)^2 . \] A tractable subfamily of this equation is completely described and shown to generate infinite classes of identities analogous to the classical Sophie Germain identity \[ x^4 + 4y^4 = (x^2 + 2y^2 + 2xy)(x^2 + 2y^2 - 2xy). \] Several Aurifeuillean factorizations arise naturally as special cases. As an application we introduce a family of infinite series derived from these identities.  This family exhibits three distinct structural regimes. First, for a discrete arithmetic locus of the parameter, the corresponding terms factor through an Aurifeuillean decomposition leading to an exact telescoping mechanism and finite rational evaluations. Second, away from this locus the same series admits natural representations in terms of special functions, including digamma functions and quadratic resolvent sums. Third, the large-parameter asymptotic expansion of the series involves odd values of the Riemann zeta function. The resulting family therefore combines an arithmetic telescoping locus with a broader analytic structure governed by special--function identities and Mellin-type asymptotics. In particular, it provides an explicit example where a Diophantine factorization gives rise to a nontrivial analytic family of series linking rational evaluations, special functions, and zeta-value asymptotics.