2012/12/29 by Romeo Meštrović, Mestrovic, Romeo
Mathematics · #05A10 #11A07 #11B65 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1301.0252
openalex publication_date 2012/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be a prime, and let k,n,m,n0 and m0 be nonnegative integers such that k≥ 1, and 0 and m0 are both less than p. K. Davis and W. Webb established that for a prime p≥ 5 the following variation of Lucas' Theorem modulo prime powers holds npk +n0 \choose mpk+m0≡np\lfloor(k-1)/3\rfloor \choose mp\lfloor(k-1)/3\rfloor n0 \choose m0 \pmodpk. In the proof the authors used their earlier result that present a generalized version of Lucas' Theorem. In this paper we present a a simple inductive proof of the above congruence. Our proof is based on a classical congruence due to Jacobsthal, and we additionally use only some well known identities for binomial coefficients. Moreover, we prove that the assertion is also true for p=2 and p=3 if in the above congruence one replace \lfloor(k-1)/3\rfloor by \lfloor k/2\rfloor, and by \lfloor (k-1)/2\rfloor, respectively. As an application, in terms of Lucas' type congruences, we obtain a new characterization of Wolstenholme primes.