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.
Solution
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).
This means that p has at least two factors, n-1 and n+1.
But, by definition, a prime number only has two distinct factors, 1 and itself.
Therefore, our assumption that there is a prime number p that is 1 less than a square number and not equal to 3 leads to a contradiction.
So, 3 is the only prime number which is 1 less than a square number.
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