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question

How many perfect squares can be written in the form $2{a^{2}}+3{b^{2}}$ with $a, b \in \mathbb{N}$?

my idea

I realised that the only solution is $(a,b)=(0,0)$

Let $a,b>0$ and we have to show that there are no perfect squares that can be written as $2{a^{2}}+3{b^{2}}$.

If $a=b \implies 5{a^{2}}$ should be a square number, but it's not, so we solved one case

If $a>b$ and $b>a$ are the cases left. i tried showing them by writing $a-b=k$ or $b-a=k$ but I get nowhere.

Sebastiano
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IONELA BUCIU
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  • @Sil Thank you so much for linking this... But I don't understand the answer from here ,,Then what happens ?$ x=3x′ , y=3y′ , z=3z′$ with $x', y', z' $being integers. What can you show and how to conclude ?" – IONELA BUCIU Jan 06 '24 at 17:42
  • If $x,y,z$ was the smallest non-zero solution, then this constructs smaller $x',y',z'$ solution, a contradiction. – Sil Jan 06 '24 at 17:47
  • @IONELABUCIU, this method of proof is basically known as proof by infinite descent. This question is among the easiest applications of this technique. – Sahaj Jan 06 '24 at 18:03

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