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Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n−1)(b^n−1)$ is not a perfect square.
[still can't solve ]

I think Zsigmondi's Theorem can help in this (Maybe).
http://en.wikipedia.org/wiki/Zsigmondy's_theorem

i am trying with LTE and infinite descense and i think i am close to a solution.
anyways thanks for the hint.

I've made a partial solve:if $(a-1)(b-1)$ is not a perfect square, then there are infinite such $n$s.But there is still an unsolved case.I've a question.How will you use infinite descent here?Eager to know.