Question #329607

. Prove that if n is a perfect square, then n + 2 is not a



perfect square.




1
Expert's answer
2022-04-18T01:52:00-0400

Let n=k2,n=k^2, where kk is some integer number.

Let's find the nearest to the nn number that is a perfect square and is greater than nn.

Obviously, this number is (k+1)2=k2+2k+1=n+2k+1(k+1)^2=k^2+2k+1=n+2k+1 and it is bigger than n+2n+2 when k1k\ge1. So, n+2\sqrt{n+2} should lie between kk and k+1k+1 and it is not an integer number.


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