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

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The discussion centers on proving that an odd prime \( p \) expressible as \( p = x^2 + 5y^2 \) must satisfy \( p \equiv 1 \) or \( 9 \mod 20 \). Participants reference basic genus theory as a foundation for the proof, while also addressing a misunderstanding regarding the interpretation of the original question. Clarifications were made about the correct modular conditions and the nature of the proof. The conversation emphasizes the importance of precise communication in mathematical discussions. Overall, the thread highlights the connection between number theory and modular arithmetic in the context of prime representation.
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If $p$ is an odd prime expressible as
$$p=x^2+5y^2$$
where $x,y$ are integers, then prove that $p \equiv 1 \text{ or }9\text{ (mod 20)}$.
 
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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 :

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

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

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

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

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

How does (2) follows from (1)?
 
Re: Primes of the form $x^2+5y^2$

mathbalarka said:
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

$\chi$ $\sigma$
 
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?
 
Re: Primes of the form $x^2+5y^2$

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

Shobhit said:
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.
 
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