Prime numbers problem - discrete math

Asked May 20 '13 at 06:00

Active May 20 '13 at 07:35

Viewed 105 times

Show that natural numbers of the form $n^2+1$ are not divisible by primes of the form $p=4k-1$.

I can't really find a place to start.

Thank you very much in advance,

Yaron.

edited May 20 '13 at 06:05

asked May 20 '13 at 06:00

user76508

1

http://math.stackexchange.com/questions/122048/1-is-a-quadratic-residue-modulo-p-if-and-only-if-p-equiv-1-pmod4 – lab bhattacharjee May 20 '13 at 06:02
hi, i can't find any similarity between the 2 questions – user76508 May 20 '13 at 06:07
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@user76508, if $p$ divides $n^2+1, -1\equiv n^2\pmod p$ – lab bhattacharjee May 20 '13 at 06:09
$x$ such that $x^2 \equiv −1 \pmod p$ if and only if $p \equiv 1 \pmod 4 \implies p \not \equiv -1( \mod 4)$, do you see that? – Inceptio May 20 '13 at 06:09
@ lab bhattacharjee got it, thanks – user76508 May 20 '13 at 06:10
@Inceptio I get that, but i don't understand how they showed that $p=1 (mod4)$. – user76508 May 20 '13 at 06:23
I don't really understand what they were trying to do – user76508 May 20 '13 at 06:31
@user76508 :
(1) $;\left(\Bbb Z/p\Bbb Z\right)^*;$ is a (cyclic) group ;

(2) $;|\left(\Bbb Z/p\Bbb Z\right)^*|=p-1;$ ;

(3) if $,x\in\left(\Bbb Z/p\Bbb Z\right)^*;$ and $,x^2=-1;$ , then $;ord(x)=4\implies 4\mid (p-1)\implies p=1\pmod 4,$
– DonAntonio May 20 '13 at 06:34
@DonAntonio why is $ord(x)=4$? – user76508 May 20 '13 at 06:41
@user76508 , because $;x^4=(x^2)^2=(-1)^2=1;$ and clearly $,x^m\neq 1;$ for natural $,m<3,$ ... – DonAntonio May 20 '13 at 06:44
@DonAntonio ok got that.
and after i now that $ord(x)=4$, why do i know that $4|p-1$?
– user76508 May 20 '13 at 06:46
@user76508, this is a basic fact of group theory. Haven't you covered that already? If you have you may want to refresh your memory with Lagrange's Theorem (google it), but if you haven't then this won't help you... – DonAntonio May 20 '13 at 06:51
@DonAntonio got it, thank you very much. I probably should refresh my memory – user76508 May 20 '13 at 06:52
@user76508: Now that you have got your solution. Write an answer for it and accept it. So that it doesn't keep popping in unanswered list of questions. – Inceptio May 20 '13 at 15:10
This should be marked as a duplicate. – Caleb Stanford May 20 '13 at 19:09

Prime numbers problem - discrete math

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