If an integer greater that 4 is a perfect square, then the immediately preceding integer is not prime.

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berggansS · Accepted Answer

Proof:  Consider integer greater than 4, which is a perfect square.  32, 42, 52, 62, 72, 82, 92, 102, 112, s˙ . or  9, 16, 25, 36, 49, 64, 81, 100, 121, s˙  Now we will show that the immediately preceding integer is not a prime.  Hence,  32 − 1=(3 − 1) (3 + 1)=2×4×1  42 − 1=(4 − 1) (4 + 1)=3×5×1  52 − 1=(5 − 1) (5 + 1)=4×6×1  ⋮  n2 − 1=(n − 1) (n + 1)=(n − 1) × (n + 1) × 1  Therefore, we can see that all numbers have two factors which are greater than one.  Further, we know that the prime number is a number that is divisible only by itself and 1. Thus, numbers that have more than two factors are not primes.  Now, we can conclude that, if an integer greater than 4 is a perfect square, then the immediatelly preceding integer is not prime.

If an integer greater that 4 is a perfect square, then the immediately preceding integer is not prime.

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