Ramanujan-Nagell Equation

The Ramanujan-Nagell equation asks for integer solutions of 2 to the n minus 7 equals x squared, and it has exactly five solutions, at n equal to 3, 4, 5, 7 and 15, corresponding to x equal to 1, 3, 5, 11 and 181. Srinivasa Ramanujan first conjectured in 1913 that these were the only solutions, Wilhelm Ljunggren proposed the same claim independently in 1943, and Trygve Nagell proved it in 1948. As an exponential diophantine equation with only finitely many solutions, it has influenced the study of related equation families, and it also implies the non-existence of perfect binary codes with minimum Hamming distance 5 or 6.