On generalized Lebesgue-Ramanujan-Nagell equations

We give a brief survey on some classical and recent results concerning the generalized Lebesgue-Ramanujan-Nagell equation. Moreover, we solve completely the equation x + 1117 = y in nonnegative integer unknowns with n ≥ 3 and gcd(x, y) = 1. 1 Generalized Ramanujan-Nagell equations Mixed polynomial-exponential equations are of classical and recent interest. One of the most famous equation of this type is the so-called RamanujanNagell equation, that is x + 7 = 2, (1) where the unknowns (x, n) are positive integers. In 1913 Ramanujan [76] conjectured that the above equation has only the solutions (x, n) ∈ {(1, 3), (3, 4), (5, 5), (11, 7), (181, 15)}. Ljunggren posed the same problem in 1943 and Nagell [70] confirmed it in 1948. His proof in English was published in 1960 (see [72]). Subsequently, Chowla,