Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc.. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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What are the bases $\beta$ such that a number with non-periodic expansion can be approximated with infinitely many numbers with periodic expansion?

Warning: This problem requires a bit of setting. Fix a finite set $A \subset \Bbb{Z}$ and consider an infinite non (ultimately) periodic sequence $\mathbf{a}=(a_i)_{i \geq 1}$ of elements of $A$ such ...
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Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains

For a non-square integer $d$ such that $d \equiv 1 \mod 4,$ we define the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}.$$ Prove that ...
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Product of Stirling Numbers of the first kind

I have been messing around with coefficients of various polynomials and was wondering if there was a way to reduce the following stuff. Let polynomial, ...