On Sum of two squares

In this discussion I will prove the following statement:

If there exists two positive integers both of which can be written as sum of two squares of integers, then there product can also be written as sum of two squares of integers

Proof :

Let two positive integers $x$ and $y$ such that $x=a^2+b^2;y=m^2+n^2;$ where all of $a,b,m$ and $n$ are integers $\Rightarrow xy=(a^2+b^2)(m^2+n^2)$ $\Rightarrow xy=a^2m^2+b^2n^2+a^2n^2+b^2m^2$ $\Rightarrow xy=(am+bn)^2\cancel{-2abmn}+(an-bm)^2\cancel{+2abmn}$ $\Rightarrow \boxed{xy=(am+bn)^2+(an-bm)^2}$

#NumberTheory

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Comments

I have a request for you, @Zakir Husain: Can you prove this:

$p_1^n + p_2^n = p_3^n$ or $p_1^n + p_2^n \neq p_3^n$

And $p_1, p_2, p_3$ are all primes...

A Former Brilliant Member - 1 year ago

@Yajat Shamji- What to prove?

Zakir Husain - 1 year ago

Read the edited comment...

A Former Brilliant Member - 1 year ago

@A Former Brilliant Member – If $n>2$ no solution will exist see Fermat's last theorem. If $n=2$ I have proved for that no solution exists for primes. If $n=1$ then, $p_1+p_2=p_3$ In the above equation, if all of $p_1,p_2,p_3$ are primes, Then $p_3>p_2;p_3>p_1$ therefore $p_3$ is greater than the $2_{nd}$ prime $\Rightarrow p_3>3$ . $\Rightarrow p_3$ is odd $\therefore$ both $p_1,p_2$ can't be odd. Therefore one of them must be an even prime $\Rightarrow p_1=2$ (Assuming $p_2>p_1$ ). And if so $2=p_3-p_2 \Rightarrow p_1$ and $p_2$ are twin primes.

And according to the twin prime conjecture (not proved yet) there are infinitely many such primes.

Zakir Husain - 1 year ago

@Zakir Husain – Your right arrow is in red - it should look like this: $\rightarrow$ - just de-capitalise the R.

A Former Brilliant Member - 1 year ago

@A Former Brilliant Member – I have edited the comment

Zakir Husain - 1 year ago

@Zakir Husain – Thanks. Also, can you prove for negative integers, the same thing as well?

And if all is done, make a note on it! I would appreciate it if I was the person behind the idea for the note!

@Zakir Husain

A Former Brilliant Member - 1 year ago

@A Former Brilliant Member – If you didn't care whether $n$ is integer or not, then just let me know

Zakir Husain - 1 year ago

@Yajat Shamji- Proof for $n<0$ ; Let $-k=n;k \in Z^+$ $p_1^n+p_2^n=p_3^n \Rightarrow \frac{1}{p_1^k}+\frac{1}{p_2^k}=\frac{1}{p_3^k}$ $\frac{p_1^k+p_2^k}{p_1^kp_2^k}=\frac{1}{p_3^k}$ ${p_1^kp_2^k}=p_3^k(p_1^k+p_2^k)$ ${p_1^k}=\frac{p_3^k(p_1^k+p_2^k)}{p_2^k} \Rightarrow p_2^k | p_3^k(p_1^2+p_2^2)$ $\Rightarrow Either :p_2^k | p_3^k \Rightarrow p_2 | p_3$ $Or : p_2^k | (p_1^k+p_2^k) \Rightarrow p_2^k | p_1^k \Rightarrow p_2 | p_1$ Both of the above statements leads us to conclude that $p_2$ will always divide either of $p_3,p_1$

$\therefore$ All of $p_1,p_2,p_3$ can't be prime

Note : $a|b$ means ' $a$ divides $b$ ' or you may translate it as $b ≡ 0 (\mod a)$

Zakir Husain - 1 year ago

Nice! Can you make a note on this and the positive proof...

A Former Brilliant Member - 1 year ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$