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(a^x-1)/(a - 1)=(b^y -1)/(b - 1) two solution in N, means a,b coprime

Source: Bulgaria NMO 2009 p1

May 28, 2019

number theoryDiophantine equationExponential equationcoprime

Problem Statement

The natural numbers

a

and

b

satisfy the inequalities

a > b > 1

. It is also known that the equation

\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}

has at least two solutions in natural numbers, when

x > 1

and

y > 1

. Prove that the numbers

a

and

b

are coprime (their greatest common divisor is

1