prove by contradiction that 3 is the only prime number which is 1 less than a square number

Question

prove by contradiction that 3 is the only prime number which is 1 less than a square number
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Solution 1

Sure, here is the proof by contradiction:

Assume that there is a prime number p that is 1 less than a square number, and p is not equal to 3.

This means that p can be written as n^2 - 1 for some integer n.

We can factor the expression n^2 - 1 into (n-1)(n+1).

So, p = (n-1)(n+1).

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