MathDB

1

Part of 2009 Bulgaria National Olympiad

Problems(1)

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

Source: Bulgaria NMO 2009 p1

5/28/2019
The natural numbers aa and bb satis fy the inequalities a>b>1a > b > 1 . It is also known that the equation ax−1a−1=by−1b−1\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1} has at least two solutions in natural numbers, when x>1x > 1 and y>1y > 1. Prove that the numbers aa and bb are coprime (their greatest common divisor is 11).
number theoryDiophantine equationExponential equationcoprime