S. Chowla; Leudesdorf's Generalization of Wolstenholme's Theorem, Journal of the London Mathematical Society, Volume s, Issue 4, 1 October , Pages. mial coefficients has a property similar to Wolstenholme's theorem, namely u(p) ≡ −1 Table 3: The first 30 values of u(n); s(n):= sign((−1)nu(n)). 2. Growth. Wolstenholme's theorem (see, e.g., [23], [7]) asserts that if p is a prime greater .. where ordp(s) is the largest power of p dividing s, and the summation is taken.

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McIntosh for all prmes p 7. For computatonal purposes ths s not much better.

Full text of "Sylvester's, Wolstenholme's, Morley's and Lehmer's Congruence Theorems Revisited"

Our computatons would be much easer f we can work wolstenholme s theorem p rather than modulo p 2. The frst congruence s a specal case of Glasher s congruence [7, p. Substtutng ths nto the frst congruence completes the proof of the theorem. There are many congruences smlar to those above nvolvng sums of lke powers of numbers n arthmetc progresson and Bernoull numbers modulo prme powers.

For an excellent source we refer the reader to a paper by Emma Lehmer [5]. For all prmes p the followng wolstenholme s theorem equvalent: A prme p s regular f and only f p does not dvde the numerators of the Bernoull numbers B 2, B 4, For all such prmes Fermat s Last Theorem s true see [6, p.

Ths does not mply, of course, that Fermat s Last Theorem s false for rregular prmes, but only that more powerful technques are requred. Ths s one of the man reasons for the wolstenholme s theorem of rregular prmes.

The Wolstenholme prmes are those rregular prmes where p dvdes the numerator of B p 3.

Johnson [0] and S. The second was found by J. We conjecture that there are nfntely many Wolstenholme wolstenholme s theorem and provde the followng heurstc argument.

Wolstenholme's Theorem

Thus p s a Wolstenholme prme f and only f W p 0 mod p. As numercal evdence suggests, we assume that the remander modulo p of W p s random.

It follows from the prme number theorem that the number of Wolstenholme prmes x s about ln ln xwhch grows very slowly to nfnty see [23, p. A smlar argument suggests that there are at most fntely many prmes p satsfyng 2p mod wolstenholme s theorem 5.

McIntosh In vew of the numercal and probablstc evdence the converse of Wolstenholme s Theorem s undoubtedly true, but a rgorous proof has not yet been obtaned. The author s grateful to J.

On the converse of Wolstenholme s Theorem

Jones for ntroducng hm to Hlbert s tenth problem and to Peter Montgomery for several of the computer programs used n the search for new Wolstenholme prmes. Monthly 48 wolstenholme s theorem[2] V. Jacobsthal, On the dvsblty of the dfference between two bnomal coeffcents, n: Den te Wolstenholme s theorem Matematkerkongress, Trondhem,[3] J.

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