Proving the Congruence of Primes of the Form x² + 5y² (mod 20)

[Shobhit]

If $p$ is an odd prime expressible as
$$p=x^2+5y^2$$
where $x,y$ are integers, then prove that $p \equiv 1 \text{ or }9\text{ (mod 20)}$.

 

[mathbalarka]

Re: Primes of the form $x^2+5y^2$

Very nice, I like this one. The proof follows immediately from basic genus theory, and I am not going to post it down here (spoiler purpose).

For beginners, I would give a hint :

Spoiler

A prime of the particular form is of the form 1, 3, 7 or 9 modulo 20

Balarka
.

 

[chisigma]

Re: Primes of the form $x^2+5y^2$

If $p$ is an odd prime expressible as
$$p=x^2+5y^2$$
where $x,y$ are integers, then prove that $p \equiv 1 \text{ or }9\text{ (mod 20)}$.

Spoiler

It must be necessarly x even and y odd or x odd and y even. In the first case You set x = 2 n and y = 2 m + 1 so that is...

$\displaystyle p = 4\ n^{2} + 20\ m^{2} + 20\ m + 5\ (1)$

... and then...

$\displaystyle p \equiv 4\ n^{2} + 5\ \text{mod}\ 20\ (2)$

Observing (2) it is evident that $\displaystyle p \equiv 1\ \text {mod}\ 20$ or $\displaystyle p \equiv 9\ \text {mod}\ 20$. The same result You obtaining supposing x odd and y even...

Kind regards

$\chi$ $\sigma$

 

[mathbalarka]

Re: Primes of the form $x^2+5y^2$

How does (2) follows from (1)?

 

[chisigma]

Re: Primes of the form $x^2+5y^2$

How does (2) follows from (1)?

I made a mistake writing $\displaystyle p \equiv 4\ n^{2}\ \text{mod}\ 20$ instead of $\displaystyle p \equiv 4\ n^{2} + 5\ \text{mod}\ 20$, so that I corrected it... very sorry! (Angry) ...

Kind regards

$\chi$ $\sigma$

 

[mathbalarka]

Re: Primes of the form $x^2+5y^2$

Ah, I see, I have interpreted the question wrong. I interpreted is as : p is an odd prime expressible as x^2 + 5y^2 if and only if p is 1 or 9 modulo 20.

But perhaps OP must have intended exactly this? Can Shobhit clarify whether he did a typo or not?

 

[Shobhit]

Re: Primes of the form $x^2+5y^2$

Ah, I see, I have interpreted the question wrong. I interpreted is as : p is an odd prime expressible as x^2 + 5y^2 if and only if p is 1 or 9 modulo 20.

But perhaps OP must have intended exactly this? Can Shobhit clarify whether he did a typo or not?

Balarka, the question is correct. You had interpreted it correctly.

 

[mathbalarka]

Re: Primes of the form $x^2+5y^2$

Balarka, the question is correct. You had interpreted it correctly.

I see, thank you for clarifying.

One can carry on with my hints on the #2 then.