This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime $p$ is Fibonacci-Wieferich is equal to $1/p$. According to our computational results and some theoretical consideratons, another form of probability can...

In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1}\equiv (p/3)3^{p-1}\left(modp²\right)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1+x+x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{p-1}{k}\right)\left(\genfrac{}{}{0pt}{}{2k}{k}\right)({(-1)}^k-{(-3)}^{-k})\equiv (p/3)(3^{p-1}-1)\left(modp³\right)$. In addition, we get some new combinatorial identities.

We establish q-analogs for four congruences involving central binomial coefficients. The q-identities necessary for this purpose are shown via the q-WZ method.

It is known that a set $H$ of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if ${dim}_{\mathscr{H}}\left(X_H\right)=0$, where $$X_H:=\left\{x=\sum _{n=1}^{\infty }\frac{x_n}{2^n}:x_n\in \{0,1\},x_n{x}_{n+h}=0\text{for}\text{all}n\ge 1,h\in H\right\}$$ and ${dim}_{\mathscr{H}}$ denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set $X_H$ by replacing $2$ with $b>2$. It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.