On the Exponential Diophantine Equation x2 + p = 2n for Primes p ≡ 7(mod8)

Abstract

We revisit the Ramanujan–Nagell type exponential Diophantine equation x²+p=2ⁿ,  x, n ∈ Z ≥ 1,  p an odd prime,  with emphasis on the congruence class p ≡ 7(mod8). Using elementary modular arguments, we show that for n ≥ 3 the congruence p ≡ 7(mod8) is a necessary condition for solvability, but not sufficient. We then propose a reproducible search methodology (over bounded exponents and odd squares) and compile a corrected set of illustrative solutions for several primes p ≡ 7(mod8) (including multiple-solution phenomena, e.g. p = 23). Finally, we establish a clean reduction showing that the variant x² + 7 = 4^m has a unique positive solution (x, m) = (3, 2), by linking it to the classical Ramanujan–Nagell equation