<s>[INST] Find all natural number(s) $n$ such that $55\mid n^2 + 3n + 1$ 
Find all natural number(s) $n$ such that $55\mid n^2 + 3n + 1$

I see that $n^2+3n+1 =n^2-2n+1+5n\equiv n^2-2n+1=(n-1)^2 \pmod{5}$
I also see that $n^2+3n+1=n^2+3n-10+11=(n-2)(n+5)+11\equiv(n-2)(n+5) \pmod{11}$
After that what?
 [/INST] There are infinitely many solutions, in particular you have the identity
$$(55t-119)^2+3(55t-119)+1=55(55t^2-235t+251)$$which shows solutions $x=55t-119$ for each integer $t$ you choose. There are maybe other solutions out of this infinity of them.
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