指数丢番图方程(5^(m)-1)/4=(p^(n)-1)/(p-1)的正整数解

Positive integer solutions of the exponential Diophantine equation (5^(m)-1)/4=(p^(n)-1)/(p-1)

为验证Ratat与Goormaghtigh关于Goormaghtigh方程仅有2组正整数解这一猜想,运用Gel′fond-Baker方法研究了一类特殊Goormaghtigh方程的可解性。设p为满足p≡1(mod 4)的素数,5是模p的原根。证明了当p>108且Pell方程u2-(5/16)(p-1)(p-5)v^(2)=1的最小解(u_(1),v_(1))满足ln[u(1)+(1/4)v_(1)√5(p-1)(p-5)]<p^(1/4)时,指数丢番图方程(5^(m)-1)/4=(p^(n)-1)/(p-1)(m>n>2)没有正整数解(m,n)。

Ratat and Goormaghtigh conjectured that the Goormaghtigh equation has only two positive integer solutions.To verify this conjecture,the solvability of a special Goormaghtigh′s equation is investigated by using the Gel′fond-Baker method.Suppose that pis a prime with p≡1(mod4)and that 5is a primitive root modulo p.It is proved that if p>108and the least solution(u_(1),v_(1))of Pell′s equation u^(2)-(5/16)(p-1)(p-5)v^(2)=1satisfiesln[u(1)+(1/4)v_(1)√5(p-1)(p-5)]<p^(1/4) then the exponential Diophantine equation(5^(m)-1)/4=(p^(n)-1)/(p-1)(m>n>2)has no positive integer solutions(m,n).