We study non-negative integer solutions (x, y, z) of the Diophantine equation (10m2 + 1)x + (15m2 − 1)y = (5m)z, where m is a positive integer. This equation is related to Terai’s conjecture, which concerns the uniqueness of integer solutions to certain exponential Diophantine equations. We provide a partial affirmation of Terai’s conjecture by proving that the equation has the unique positive integer solution (x, y, z) = (1, 1, 2) for all m > 1. The proof relies on the well-known primitive divisor theorem and several results from the theory of exponential Diophantine equations.