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 = 4m has a unique positive solution (x, m) = (3, 2), by linking it to the classical Ramanujan–Nagell equation
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Analytic Number Theory Research
Advanced Mathematical Identities
Publication details
- Journal / source
- International Journal of Global Sustainable Research
- Volume
- 4
- Issue
- 2
- Pages
- 231–236
- Publication date
- 5 March 2026
- Language
- EN