Solve Herstein's Abstract Algebra Problem: Can u = 4n+3 be Written as a^2 + b^2?

[JasonRox]

This isn't homework, but I'll post it anyways because I'd like to know.

It's from Herstein's Abstract Algebra.

Show that if u = 4n + 3, where [itex]n\inN[/itex], then you can not write u in the from u = a^2 + b^2, where [itex]a,b\inN[/itex].

I feel silly for asking this, but I'm curious to know.

The one thing I do, which is obvious is that if a is odd, then b is even because u is odd. But I don't think you need this fact to solve it.

Any directions?

Please do not post solutions!

 

[VietDao29]

This isn't homework, but I'll post it anyways because I'd like to know.

It's from Herstein's Abstract Algebra.

Show that if u = 4n + 3, where [itex]n\inN[/itex], then you can not write u in the from u = a^2 + b^2, where [itex]a,b\inN[/itex].

I feel silly for asking this, but I'm curious to know.

The one thing I do, which is obvious is that if a is odd, then b is even because u is odd. But I don't think you need this fact to solve it.

Any directions?

Please do not post solutions!

Okay, I'll give you a hint:
So one of a, and b must be odd, and the other is an even number, right?
So let a = 2k, b = 2x + 1 (k, x are all integers).
Now what's a² + b²? If you divide a² + b² by 4, what's the remainder?
You can take it from here, right? :)

 

[JasonRox]

Okay, I'll give you a hint:
So one of a, and b must be odd, and the other is an even number, right?
So let a = 2k, b = 2x + 1 (k, x are all integers).
Now what's a² + b²? If you divide a² + b² by 4, what's the remainder?
You can take it from here, right? :)

That's exactly what I did!

I knew something wasn't right when I was looking at it.

I'll give it another shot thanks.