integers

**mpryal** · May 1st 2009, 05:48 AM

Show that no integer u=4n+3 can be written as u = a^2 + b^2 where a,b are integers.

**Moo** · May 1st 2009, 05:59 AM

Hello,

Originally Posted by mpryal

Show that no integer u=4n+3 can be written as u = a^2 + b^2 where a,b are integers.

Fact : For any integer m, $m^2 \equiv 0 \text{ or } 1 (\bmod 4)$

Proof :
A number m can only be even or odd.
If it's even, it can be written m=2k, in which case, $m^2=4k^2 \equiv 0 (\bmod 4)$
If it's odd, it can be written $m=2k+1$ , in which case, $m^2=4k^2+4k+1 \equiv 1 (\bmod 4)$

Now, is it possible that $u=a^2+b^2 \equiv 3 (\bmod 4)$ when $a^2$ and $b^2$ can only $\equiv 0 \text{ or } 1 (\bmod 4)$ ?

Thread: integers

LinkBack

Thread Tools

Display

integers

Similar Math Help Forum Discussions

The number of integers are infinite and the integers are in infinite set

Circular numbers as opposed to integers and Gaussian integers?

Matrix of integers whose inverse is full of integers

Choosing Non-Consecutive Integers from a list of N Integers

If r, q are rationals, and r+q, r*q are integers, are r, q both integers?

Search Tags