A binomial sum related to Wolstenholme's theorem

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• 作者：Marc Chamberl ; Karl Dilcher
• 关键词：Binomial sums ; Binomial coefficients ; Wolstenholme's theorem ; Congruences ; Wilf– ; Zeilberger method
• 刊名：Journal of Number Theory
• 出版年：2009
• 出版时间：November 2009
• 年：2009
• 卷：129
• 期：11
• 页码：2659-2672
• 全文大小：190 K

文摘

A certain alternating sum u(n) of n+1 products of two binomial coefficients has a property similar to Wolstenholme's theorem, namely for all primes p5. However, this congruence also holds for certain composite integers p which appear to always have exactly two prime divisors, one of which is always 2 or 5. This phenomenon will be partly explained and the composites in question will be characterized. We also study the sequence u(n) in greater detail, especially its growth and its sign distribution.