Elementary # Theory

**felixmcgrady** · September 30th 2008, 06:56 PM

Hey, guys, this might be a stupid question, but

How do you prove that

4K+3 can not be a perfect square, if K is any integer.

Thanks a lot .

**icemanfan** · September 30th 2008, 07:06 PM

All integers are of four different forms: 4x, 4x + 1, 4x + 2, and 4x + 3 for some integer x. Since squares of even numbers are even, 4x + 3 must be the square of an odd number, if it would be a square. All odd numbers are either of the form 4x + 1 or 4x + 3. Now show that $(4x + 1)^2$ and $(4x + 3)^2$ are both of the form 4x + 1, and thus 4x + 3 cannot be the square of any odd integer, and hence cannot be the square of any integer.

**Moo** · October 4th 2008, 05:37 AM

Hello,

More simply, a number is either 2n or either 2n+1.

$(2n)^2=4n^2=4n' \neq 4k+3$

$(2n+1)^2=4n^2+4n+1=4(n^2+n)+1=4n'+1 \neq 4k+3$

Thread: Elementary # Theory

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