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

allhvasstH

allhvasstH

Answered question

2020-11-10

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

Answer & Explanation

berggansS

berggansS

Skilled2020-11-11Added 91 answers

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.

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