arXiv subjects
Publications and source records attributed to Konstantinos Gaitanas.
Let $\mathcal{N}[k]$ be the multiset containing the $\binom{n-1}{k}$ products of $k$-subsets of $\{1,\ldots, n-1\}$. We show that if $n\geq (2c+3)^2$, then \begin{gather*}\left((-1)^c+\sum_{M\in \mathcal{N}[n-1-c]}M\right)\cdot(c+1)\equiv 0\pmod{n},\end{gather*} if and only if $n=(c+1)p$, where $p$ is prime. This provides a combinatorial extension of Wilson's Theorem, which is the special case where $c=0$.
In this paper, we provide some novel results concerning the behavior of $\frac{U_{kn}}{U_k}$ modulo ${U_n}$, where $(U_n)_{n\in\mathbb{N}}$ is the Lucas sequence of the first kind. As a consequence, we obtain some primality criteria which do not seem to appear in the literature.
In this note, we provide some results concerning the structure of a set $A\subseteq \mathbb{Z}_n^{\times}$, which has non-empty subset sums equally distributed modulo $n$. Here, $\mathbb{Z}_n^{\times}$ denotes the set which contains all the invertible elements of the ring $\mathbb{Z}_n$. In particular, we prove that if $n=q$ is a power of an odd prime, then $A$ is a union of sets of the form $\{ a\cdot(\pm2^i)\}$. Additionally, we count the number of subsets of $\mathbb{Z}_q^{\times}$ with non-empty subset sums equally distributed modulo $q$.
In the current note, we present a new, short proof of the famous AM-GM-HM inequality using only induction and basic calculus.
We consider two problems regarding some divisibility properties of the subset sums of a set $A\subseteq \{1, 2, \ldots ,n\}$. At the beginning, we study the cardinality of $A$ which has the following property: For every $d\le n$ there is a non empty set $A_d\subseteq A$ such that the sum of the elements of $A_d$ is a multiple of $d$. Next, we turn our attention to another problem: If all subset sums of $A$ form a multiple free-sequence, what can we say about the structure of $A$? We give some asymptotics for the first problem and improve some already existing results for the second one.
Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a quartic polynomial for which the product of it's first consecutive values is infinitely often a perfect square.