Related papers: A search for Wilson primes

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

Difficulty of calculation of discrete logarithm for any arbitrary Field is the basis for security of several popular cryptographic solutions. Pohlig-Hellman method is a popular choice to calculate discrete logarithm in finite field $F_p^*$.…

In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$…

We develop a simple $O((\log n)^2)$ test as an extension of Proth's test for the primality for $p2^n+1$, $p>2^n$. This allows for the determination of large, non-Sierpinski primes $p$ and the smallest $n$ such that $p2^n+1$ is prime. If $p$…

A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…

The Ap\'ery polynomials are given by $$A_n(x)=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2x^k\ \ (n=0,1,2,\ldots).$$ (Those $A_n=A_n(1)$ are Ap\'ery numbers.) Let $p$ be an odd prime. We show that…

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory…

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le…

In this note, we review some facts about polynomials representing functions modulo primes p. In addition we prove that the polynomial f(x) = x^{p-2} + x^{p-3} + ... + x^3 + x^2 + 2x + 1 represents the transposition (0 1) modulo p, that is,…

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original…

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…

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to…

Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. In this note we…

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…

We derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$, within a sufficiently large degree range: $4/p^7$. As an application, we classify knot Jones polynomials modulo two of span up to eight.

Many large arithmetic computations rely on tables of all primes less than $n$. For example, the fastest algorithms for computing $n!$ takes time $O(M(n\log n) + P(n))$, where $M(n)$ is the time to multiply two $n$-bit numbers, and $P(n)$ is…

The classic way of computing a $k$-universal hash function is to use a random degree-$(k-1)$ polynomial over a prime field $\mathbb Z_p$. For a fast computation of the polynomial, the prime $p$ is often chosen as a Mersenne prime $p=2^b-1$.…

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an…

The polynomials of degree $\frac{p-1}{2}$ of range sum $p$ was determined in {\tt arXiv:2311.06136 [math.NT]} for large enough primes. We extend this result by reducing the lower bound for the primes to $23$ by introducing a new and…

Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\…