Andrew and Blair are bored in class and decide to play a game. They pick a pair \((a, b)\) with \(1 \le a, b \le 100\). Andrew says the next number in the geometric series that begins with \(a, b\) and Blair says the next number in the arithmetic series that begins with \(a, b\). For how many pairs \((a, b)\) is Andrew's number minus Blair's number a positive perfect square?

Let's think step by step and output the final answer within \boxed{}.
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