A search for Fibonacci-Wieferich and Wolstenholme primes期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

A search for Fibonacci-Wieferich and Wolstenholme primes

Authors:	Richard J McIntosh Eric L Roettger

Institution:	Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 ; Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Abstract:	A prime is called a Fibonacci-Wieferich prime if $F_{p-({p\over5})}\equiv 0\pmod{p^2}$ , where is the th Fibonacci number. We report that there exist no such primes $p<2\times10^{14}$ . A prime is called a Wolstenholme prime if ${2p-1\choose p-1}\equiv 1\pmod {p^4}$ . To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes . Wolstenholme, in 1862, proved that ${2p-1\choose p-1}\equiv 1\pmod {p^3}$ for all primes $p\ge 5$ . It is estimated by a heuristic argument that the ``probability' that is Fibonacci-Wieferich (independently: that is Wolstenholme) is about . We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient $F_{p-({p\over5})}/p$ modulo .

Keywords:	Fibonacci number Wieferich prime Wall-Sun-Sun prime Wolstenholme prime

	点击此处可从《Mathematics of Computation》浏览原始摘要信息
	点击此处可从《Mathematics of Computation》下载免费的PDF全文