Ramanujan-Nagell equation is:
2n - 7 = x2
with positive solutions:
(n, x) = (3,1), (4, 3), (5, 5), (7, 11) and (15, 181).
References to the applications:
H. S. Shapiro and D. L. Slotnick, "On the mathematical theory of error-correcting codes", IBM J. Res. Develop. 3 (1959), 25-34. D. G. Mead, "The equation of Ramanujan-Nagell and [y2]", Proc. of Amer. Math. Soc. 41 No. 2 (1973), 333-341.