Related papers: A Note on Permutation Binomials and Trinomials ove…

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd}…

For an odd positive integer $n\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b (a,b\in \mathbb Z)$ are irreducible and their…

Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all…

An infinite permutation is a linear order on the set N. We study the properties of infinite permutations generated by fixed points of some uniform binary morphisms, and find the formula for their complexity.

Natural numbers which are nontrivial multiples of some permutation of their base-$b$ digit representations are called permutiples. Specific cases include numbers which are multiples of cyclic permutations (cyclic numbers) and reversals of…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

We give all non splitting bi-unitary perfect polynomials over the prime field of two elements, which have only Mersenne polynomials as odd irreducible divisors.

Let $q=2^m.$ In a recent paper \cite{Zhang3}, Zhang and Zheng investigated several classes of permutation pentanomials of the form $\epsilon_0x^{d_0}+L(\epsilon_{1}x^{d_1}+\epsilon_{2}x^{d_2})$ over ${\mathbb F}_{q^3}~(d_0=1,2,4)$ from some…

We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small…

Let $\mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with…

We give a new bound on colinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

We obtain several estimates for bilinear form with exponential sums with binomials $mx^k + nx^\ell$. In particular we show the existence of nontrivial cancellations between such sums when the coefficients $m$ and $n$ vary over rather sparse…

In this note we continue the analysis of permutations that avoid substrings j(j+k), 1 <= j <= n-k, k < n, as well as substrings j(j+k) (mod n), 1 <= j <= n. In the first case the number of such permutations can be obtained from recursions…

In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0,…

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…

Let $p$ be a prime and $n$ be a positive integer. Let $f_b(X)=X+(X^p-X+b)^{-1}$, where $b\in\Bbb F_{p^n}$ is such that $\text{Tr}_{p^n/p}(b)\ne 0$. In 2008, Yuan et al. \cite{Yuan-Ding-Wang-Pieprzyk-FFA-2008} showed that for $p=2,3$, $f_b$…

We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1, d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=\delta n+(\delta+2).$$ We prove the similar result for $\varepsilon=\delta-2$ and…

Let n,p,k be three positive integers. We prove that the numbers binomial (n,k) 3F2 (1-k, -p, p-n ; 1, 1-n ; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers,…