关于丢番图方程|3~x-2~y|=p(英文)

Unique Solution to The Diophantine Equation |3~x-2~y|=p

设p为素数,x,y正整数.在文献中,周科证明了p=41,43,53,59,67,71,83,87,97时,方程|3x-2y|=p没有解;证明了方程在p=5,7,13,23时有超过一组的解并给出了所有解,该文证明了当p=31,37,47,61,73,79时,该方程有唯一解,并给出了相应解.

Let p be prime number, x and y be positive integral numbers, the propositions that when p = 41,43,53, 59,67,71, 83,87,97, the Diophantine equation ｜ 3^x -2^y ｜ = p has no integral solution ,and the equation has more than a group of solution when p = 5,7,13,23 that has been proved by Zhou Ke. In this paper, we study the unique solution to this equation in case p = 31,37,47,61,73,79, also all of those integral solutions will be given.