<s>[INST] Is there $n\in\mathbb N$ for which $3n+1$ and $2n+1$ are perfect squares and $5n+3$ is a prime? 
Is there  $n\in\mathbb N$ for which $3n+1$ and $2n+1$ are perfect squares and $5n+3$ is prime?

My trying:
We know that $n=3n+1-(2n+1)$ is not prime.
But how it can help for $5n+3$? 
$\mathbb N=\{1,2,...\}$.
Thank you!   
 [/INST] Lets take $2n+1=a^2$ and $3n+1=b^2$,WLOG assume $a,b>0$ then $4a^2-b^2=(2a-b)(2a+b)=5n+3$ this is prime only if $2a-b=1$ and $2a+b$ is prime.
Take $b=2a-1$ then $$3n+1=4a^2-4a+1\\3n=4(2n+1)-4\sqrt{2n+1}\\5n+4=4\sqrt{2n+1}\\25n^2+40n+16=16(2n+1)\\25n^2+8n=0$$so either $n=0$ or $n=-\frac{8}{25}$,but $0,-\frac{8}{25}\not\in \Bbb{N}$
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