ON THE SOLUTIONS OF THE EXPONENTIAL DIOPHANTINE EQUATION a~x + b~y = (m~2 + 1)~z

摘要

In this paper, we consider the Diophantine equation a~x + b~y = c~z. Combining Laurent's result on lower bounds for linear forms in two logarithms, Bugeaud's result on upper bounds for the p-adic logarithms, and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers, we obtain a sharper computable upper bound for the number of positive integer solutions of the equation |v_r|~x + |v_r|~y = c~z, where v_r~2 + u_r~2 = (m~2 + 1)~r and r is an odd number. Moreover, if r is a prime and r ≡ 5 (mod 8) or r ≡ 19 (mod 24), m ≥ 2/πr, we prove that the equation has only the solution (x, y, z) = (2, 2, r).