# How do you prove that there is no perfect square of the form 4k+3?

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• Aug 1st 2012, 12:27 AM
omegatidus
How do you prove that there is no perfect square of the form 4k+3?
Im stuck on this homework problem that im not sure if i am doing right and it is due tommorow:
Prove that there is no perfect square of the form 4k+3?

Case 0:
k is even
k=2n for some n in Z
4(2n)+3=8n+3 which is odd so there is no perfect square if k is even

Case 1:
k is odd
k=2n+1 for some n in Z
4(2n+1)+3 = 8n+7 which is not of the form 4k+3 there for there is no perfect square
• Aug 1st 2012, 02:16 AM
a tutor
Re: How do you prove that there is no perfect square of the form 4k+3?
I think you're a little confused.

Squares of even numbers are even so 4k+3, which is odd of course, is not the square of an even number.

Squares of odd numbers are of the form (2m+1)^2 = 4m^2+4m+1 = 4(m^2+m) + 1 which is of the form 4k+1.
• Aug 1st 2012, 03:26 AM
omegatidus
Re: How do you prove that there is no perfect square of the form 4k+3?
Hahah i feel like such a derp!

THANK YOU xxxx SPACIBAAAAAAAA !
• Aug 1st 2012, 07:50 AM
richard1234
Re: How do you prove that there is no perfect square of the form 4k+3?
Note: this solution's similar to a tutor's solution, but uses mods instead. Note that n must be odd. We have two cases:

$n \equiv 1 (\mod 4)$. Then $n^2 \equiv 1 (\mod 4)$

$n \equiv 3 (\mod 4)$. Then $n^2 \equiv 1 (\mod 4)$.

In either of these cases $n^2 \not \equiv 3 (\mod 4)$, that is, it cannot be written in the form 4k+3, k integer. Hence we're done.