幂比较法在不定方程中的应用

Application of the Power of Comparative Method in Indefinite Equations

利用幂比较法证明了:①当a为正偶数、b为正奇数时,不定方程a^x-b^y=1最多有1组正整数解(x,y);②方程x^y-(x-1)^z=1仅有正整数解(x,y,z)=(1,s,t),(2,1,t),(r,1,1)和(3,2,3),其中r,s,t为任意正整数且r≥3.同时推出不定方程2^x-3^y=1仅有正整数解(x,y)=(2,1),不定方程2018^x-2019^y=1无正整数解以及不定方程3^x-2^y=1仅有正整数解(x,y)=(1,1),(2,3).

It has been proven by the power of comparative method that:① if a is positive even and b is positive odd, then the equation a^x-b^y =1 has at most one positive integer solution;② the equation x^y-(x-1)^ z =1 has only the positive integer solutions (x,y,z)=(1,s,t),(2,1,t),(r ,1,1) and (3,2,3),where r , s and t are arbitraty positive integers, with r ≥3. In addition, the equation 2^x-3^y =1 has only positive integer solution (x,y )=(2,1), the equation 2 018^x-2 019^y =1 has no positive integer solution, and the equation 3^x-2^y =1 has only positive integer solutions ( x,y )=(1,1),(2,3).