p is of the form m^2+2n^2

[shehab ahmed]

if p is a prime and $p^2$ can be written in the form $m^2 + 2n^2$ for positive integer m and n,then prove that p can also be written in the same form.

[*Mahi*]

Why don't you use latex? http://www.matholympiad.org.bd/forum/vi ... p?f=9&t=54 It is easy and makes everything a hundred times clear.
And as for the problem:
Hint:

I believe you had reached $p^2-m^2=8k^2$. Then notice the gcd of $(p+m),(p-m)$ equals 2. So divide it as $2i^2.2^{2x}j^2$ , and do what is needed.

[shehab ahmed]

আমি মোবাইল দিয়ে ঢুকি।এখানে ল্যাটেক্স কীভাবে ব্যবহার করতে হয় তা আমার জানা নেই।

[*Mahi*]

http://www.matholympiad.org.bd/forum/vi ... p?f=9&t=54 এই থ্রেডে সব দেয়া আছে। মোবাইল থেকে লেখার সময় কেবল equation এর দুই পাশে দুইটা $ চিহ্ন দিলেই হবে।

[afif mansib ch]

let,\[p^2=a^2+2b^2\]
where a is in the form\[m^2-2n^2\]and b is in the form \[2mn\]
so \[p=m^2+2n^2\]

[*Mahi*]

let,\[p^2=a^2+2b^2\]
where a is in the form\[m^2-2n^2\]and b is in the form \[2mn\]
so \[p=m^2+2n^2\]

Then you have to prove for all p of the form $a^2+2b^2$, a and b has the given form.

[afif mansib ch]

that would be just the reverse process,i think.

[shehab ahmed]

তুমি যে a,b এর জায়গায় m,n দিয়ে রাশি বানিয়ে প্রতিস্থাপিত করলে তোমার কাছে প্রমাণ কী যে তুমি এমন পূর্ণসংখ্যা m,n পাবে?

[Masum]

if p is a prime and $p^2$ can be written in the form $m^2 + 2n^2$ for positive integer m and n,then prove that p can also be written in the same form.

If you have done the proof of all solutions to \[a^2+b^2=c^2\]
you may use the same technique with \[p^2=a^2+2b^2\]
with the additional information that $p$ is a prime.