A Complete Resolution of the Exponential Diophantine Equation

This paper determines all nonnegative integer solutions of the exponential Diophantine equation z^2=277665⋅17^6+34^n. After an elementary branch decomposition, the even-exponent branch is reduced to a finite factorization problem. The odd-exponent branch is transformed into the generalized Pell equation X^2−34Y^2=277665, subject to the additional restriction Y=8⋅34^k. A fundamental-domain argument produces exactly eight Pell orbits, whose normalized Y-coordinates form nondegenerate binary recurrence sequences. An explicit lower bound for a three-logarithm form, followed by a corrected Dujella–Pethő continued-fraction reduction and certified numerical calculations, reduces the recurrence index to a finite terminal range. An exact check then completes the classification. The complete set of nonnegative integer solutions is (n,z)=(6,2589151),(7,2598977),(8,2913409),(10,45509119). The deposit includes the manuscript, LaTeX source, and a reproducible Python verification script for the finite computational certificates.