Wolstenholmes sats

Inom matematiken är Wolstenholmes sats ett resultat som säger att för alla primtal p > 3 gäller kongruensen:

${\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.}$

En ekvivalent formulering är kongruensen:

${\displaystyle {ap \choose bp}\equiv {a \choose b}{\pmod {p^{3}}}.}$

Satsen bevisades av Joseph Wolstenholme 1862.

Referenser

Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia, Wolstenholme's theorem, 27 januari 2014.

Källor

• Babbage, C. (1819), ”Demonstration of a theorem relating to prime numbers”, The Edinburgh philosophical journal 1: 46–49, http://books.google.com/books?id=KrA-AAAAYAAJ&pg=PA46 
• Wolstenholme, J. (1862), ”On certain properties of prime numbers”, The Quarterly Journal of Pure and Applied Mathematics 5: 35–39, http://books.google.com/books?id=vL0KAAAAIAAJ&pg=PA35 
• Glaisher, J.W.L. (1900), ”Congruences relating to the sums of products of the first n numbers and to other sums of products”, The Quarterly Journal of Pure and Applied Mathematics 31: 1–35 
• Glaisher, J.W.L. (1900), ”On the residues of the sums of products of the first p−1 numbers, and their powers, to modulus p² or p³”, The Quarterly Journal of Pure and Applied Mathematics 31: 321–353 
• McIntosh, R. J. (1995), ”On the converse of Wolstenholme's theorem”, Acta Arithmetica 71 (4): 381–389, http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf 
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862—2012).