一类指数丢番图方程的解数

设 a, b , c, k 是适合 a + b = ck, gcd( a, b) = 1, c∈ { 1, 2, 4} , k > 1且 k 在c = 1或 2 时为奇数的正整数;又设ε= ( a + - b ) / c,ε = ( a - - b ) / c. 证明了:当( a, b, c, k )≠( 1, 7, 4, 2) 或( 3, 5, 4, 2) 时,至多有1 个大于 1的正奇数 n 适合 (εⁿ-εⁿ) / (ε-ε) = 1,而且如此的 n 必为满足n < 1+ ( 2logπ) / log k + 2 563. 43( 1+ ( 21. 96π) / log k )的奇素数.

The Number of Solutions for a Class of Exponential Diophantine Eq uations

Let a, b, c, k be posi tive integers such taht a+ b= ck , gcd( a, b ) = 1, c ∈ { 1, 2, 4} , k> 1 and k is odd if c= 1 or 2. Further let  ε= ( a+ - b ) / c and  ε= ( a- - b ) / c. In this paper we prove that if ( a, b , c,k )≠( 1, 7, 4, 2 ) or ( 3, 5, 4, 2 ) , then there exist at mos t one positive odd integer n satisfying( ε ⁿ- ε ⁿ) / ( ε- ε) = 1 with n> 1.Moreover , such n must be an odd prime satisfying n< 1+ ( 2log π) / log k+ 2563. 43( 1+ 21. 96π/ log k) .