Recognizing the Square Numbers!

\[(2^n -1), \ \ (3^{2n}-1), \ \ (4^{3n}-1), \ \ (5^{4n}-1),\ \ (6^{5n}-1), \ \ (7^{6n}-1),\ \ (8^{7n}-1),\ \ (9^{8n}-1)\]

In Calvin's number set, there are the numbers as listed above for each natural number \(n\), and there are no other numbers in the set. How many square numbers does this set contain?

Note : The set of Natural Numbers doesn't contain the number \(0\).

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