If n > 3 and (n + 1) is a square, is there any n that is a prime?

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I am looking at properties of squares and came about this property. I am investigating the difference of squares in relation to primes.

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Does there exist a prime of the form $a^2-1$ when $a^2-1>3$?

No, because $$a^2-1=(a-1)(a+1)$$ and both $a-1$ and $a+1$ are non-trivial factors.