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I graduated in mathematics sometime in a past, now I work as a software engineer.
Some interesting topics:
$x^j+x^k+2$ is irreducible over $\mathbb{Q} \iff \nu_2(j) \neq \nu_2(k)$
$x^m+y^n$ is irreducible over $\mathbb{Q} \iff \gcd(m,n)=2^k$
If polynomials $P(x)=Q(y)$ for infinitely many integers $x,y$, then $P(x)=Q(R(x))$
If $f(x)=a_0+a_1x+\dots+a_nx^n$ then $\gcd(f(0),f(1),\dots)$ divides $\gcd(a_0,\dots,a_n)\cdot n!$
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