On a conjecture concerning the exponential Diophantine equation (an2+1)x + (bn2-1)y = (cn)z

Let a, b, c, and n be positive integers such that a + b = c(2), 2 <does not divide> c and n > 1. In this paper, we prove that if gcd(c, n) = 1 and n >= 117.14c, then the equation (an(2) + 1)(x) + (bn(2) - 1)(y) = (cn)(z) has only the positive integer solution (x, y, z) = (1, 1, 2) under the assumption gcd(an(2) + 1, bn(2) - 1) = 1. Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer n, if (a, b, c) = (12, 13, 5), (18, 7, 5), or (44, 5, 7), then this equation has only the solution (x, y, z) = (1, 1, 2). This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently.