A left quasigroup $(Q,q)$ of order $2^w$ that can be represented as a vector of Boolean functions of degree 2 is called a left multivariate quadratic quasigroup (LMQQ). For a given LMQQ there exists a left parastrophe operation $q_{\setminus }$ defined by: $q_{\setminus }(u,v)=w\iff q(u,w)=v$ that also defines a left multivariate quasigroup. However, in general, $(Q,q_{\setminus })$ is not quadratic. Even more, representing it in a symbolic form may require exponential time and space. In this work we investigate the problem of finding a subclass of LMQQs whose left parastrophe...

In this note we generalize some results from finite fields to Galois rings which are finite extensions of the ring Zpm of integers modulo pm where p is a prime and m ≥ 1.

We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences ${\sum }_{i=1}^k{a}_i{x}_i\equiv b(modn)$. In particular, we obtain explicit expressions for the number of solutions, where $x_i$’s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1\ge \cdots \ge x_k$ or with strict order restrictions $x_1>\cdots >x_k$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan...

Let $a_{d-1},\cdots ,a_0\in \mathbb{Z}$, where $d\in \mathbb{N}$ and $a_0\ne 0$, and let $X={\left(x_n\right)}_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}{x}_{n+d-1}+\cdots +a_0{x}_n$ for $n=1,2,3,\cdots$. We show that there are a prime number $p$ and $d$ integers $x_1,\cdots ,x_d$ such that no element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p\ge 3$ and $d$ integers $x_1,\cdots ,x_d$ such that every element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\cdots ,p-s-1\}$.