Wilson's Theorem for Finite Fields

Wilson's and Wolstenholme's Theorems

We provide a proof of Wilson's Theorem and Wolstenholme's Theorem based on a direct approach by Lagrange requiring only basic properties of the primes and the Binomial theorem. The goal is to show how similar the two theorems are by providing the easiest proof possible in a single unified argument.

q-Analogue of Wilson's theorem

We give q-analogues of Wilson's theorem for the primes congruent 1 and 3 modulo 4 respectively. And q-analogues of two congruences due to Mordell and Chowla are also established.

On Some Generalizations of Fermat's, Lucas's and Wilson's Theorems

We use actions by finite cyclic groups to derive generalizations of three classical theorems from elementary number theory.

Wilson's theorem

We show that there are four possibilities for the product of all elements in the multiplicative group of a quotient of the ring of integers in a number field, and give precise conditions for each of the possibilities to occur. This generalisation of Wilson's theorem turns out to have been first discovered by M. Laššák (2000), but our proof is simpler and more direct.

威尔逊定理模更高素数幂 II：伯努利数与伯努利多项式

Wilson's theorem modulo higher prime powers II: Bernoulli numbers and polynomials

该研究将作者此前用费马商幂和超同余描述威尔逊定理与威尔逊商的成果，转化为幂和与伯努利数的同余式，给出更简洁的证明，还算出了模p⁴的威尔逊商等结果并可推广到更高素数幂。

Stirling Functions and a Generalization of Wilson's Theorem

For positive integers m and n, denote S(m,n) as the associated Stirling number of the second kind and let z be a complex variable. In this paper, we introduce the Stirling functions S(m,n,z) which satisfy S(m,n,z) = S(m,n) for any z which lies in the zero set of a certain polynomial P(m,n,z). For all real z, the solutions of S(m,n,z) = S(m,n) are computed and all real roots of the polynomial P(m,n,z) are shown to be simple. Applying the properties of the Stirling functions, we investigate the divisibility of the numbers S(m,n) and then generalize Wilson's Theorem.

Extensions of Wilson's lemma and the Ax-Katz theorem

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 present a useful lemma on polynomials modulo prime powers, and applied this lemma to reprove the Ax-Katz theorem on solutions of congruences modulo p and deduce various results on codewords in p-ary linear codes with weights. In light of the recent generalizations of Fleck's congruence given by D. Wan, and D. M. Davis and Z. W. Sun, we obtain new extensions of Wil…

Around Wilson's theorem

We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x in the Lebesgue measure s(n,x) is equivalent to n/2 when n tends to infinity, while for almost all x in the Baire sense, 1/2 is a limit point of the ratio of s(n,x) to the number of primes less or equal to n.

威尔逊定理模更高素数幂 III：模 $p^6$ 与 $p^7$ 的情形

Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$

本研究拓展前期工作，算出更高素数幂下的威尔逊商及对应阶乘同余，确定费马商幂和模性质，揭示相关p进系数规律并验证原始威尔逊同余的适配性。

New Wilson-like theorems arising from Dickson polynomials

Wilson's Theorem states that the product of all nonzero elements of a finite field ${\mathbb F}_q$ is $-1$. In this article, we define some natural subsets $S \subset {\mathbb F}_q^\times$ and find formulas for the product of the elements of $S$, denoted $\prod S$. These new formulas are appealing for the simple, natural description of the sets $S$, and for the simplicity of the product. An example is $\prod\left\{ a \in {\mathbb F}_q^\times : \text{$1-a$ and $3+a$ are nonsquares} \right\} = 2$ if $q \equiv \pm 1 \pmod{12}$, or $-1$ otherwise.

Wilson's theorem modulo p^2 derived from Faulhaber polynomials

First, we present a new proof of Glaisher's formula dating from 1900 and concerning Wilson's theorem modulo p^2. Our proof uses p-adic numbers and Faulhaber's formula for the sums of powers (17th century), as well as more recent results on Faulhaber's coefficients obtained by Gessel and Viennot. Second, by using our method, we find a simpler proof than Sun's proof regarding a formula for (p-1)! modulo p^3, and one that can be generalized to higher powers of p. Third, we can derive from our method a way to compute the Stirling numbers modulo p^3, thus improving Glaisher and Sun's own results fr…