On The Diophantine Equation (a~n-1) (b~n-1)=x~2

In this dissertation, let N be the set of all positive integers. Given a, b∈N, thediopha~ntine equation (a~n -1)(b~n -1) = x~2 attracts ma~ny mathematicia~ns such as:L. Szalay, L. Hajdu, J.H.E. Cohn, F. Luca, P.G. Walsh a~nd Mao-Hua Le, a~nd soon. In 2002, F. Luca a~nd P.G. Walsh had resolved almost all the solutions of thediopha~ntine equation (a~n - 1)(b~n - 1) = x~2 in the ra~nge 2≤b < a≤100 withsixty-nine exceptions. Using the knowledge of quadratic residue, congruence, a~ndthe Legendre symbol, we continue this work.The main results of this thesis are divided into two parts. In the first partwe study five of the sixty-nine exceptions in [F. Luca a~nd P.G. Walsh, J. NumberTheory 96(2002), 152-173]. That is, we consider the diopha~ntine equation (a~n -1)(b~n - 1) = x~2 with (a, b) = (13, 4), (28, 13), (19, 9), (33, 9), (33, 3). In details, weprove that the equation (4~n-1)(13~n-1) = x~2 has only the solution: n = 1, x = 6 inpositive integers n a~nd x; the equation (13n-1)(28n-1) = x~2 has only the solution:n = 1, x = 18 in positive integers n a~nd x; if the equation (9n - 1)(19n - 1) = x~2has solutions in positive integers n a~nd x, then n≡1 (mod 900); if the equation(9n - 1)(33n - 1) = x~2 has solutions in positive integers n a~nd x, then n≡1(mod 13440); if the equation (3~n - 1)(33~n - 1) = x~2 has solutions in positiveintegers n a~nd x, then n≡1 (mod 3600).In the second part, we investigate diopha~ntine equation (a~n-1)(b~n-1) = x~2 forsome kinds of (a, b). In details, we prove that: if a≡3, 19, 67, 83, 131, 147, 171, 179(mod 200), then the equation (2n - 1)(a~n - 1) = x~2 has no solutions in positiveintegers with n≥2; if a≡5, 85 (mod 200) a~nd a - 1 is a perfect square, then theequation (2n - 1)(a~n - 1) = x~2 has no solutions in positive integers with n≥2.