A note on the Diophantine equation x² + qm = c2n.

Let q be an odd prime. Let c > 1 and t be positive integers such that qt + 1 = 2c². Using elementary method and a result due to Ljunggren concerning the Diophantine equation xn-1/x-1 = y², we show that the Diophantine equation x² + qm = c2n has the only positive integer solution (x, m, n) = (c² - 1, t, 2). As applications of this result some new results on the Diophantine equation x² + qm = cn and the Diophantine equation x² + (2c - 1)m = cn are obtained. In particular, we prove that Terai's conjecture is true for c = 12,24. Combining this result with Terai's results we conclude that Terai's conjecture is true for 2 ≥ c ≥ 30 [ABSTRACT FROM AUTHOR]