Editing Wolstenholme's theorem

{{Short description|Result in number theory}}
In [[mathematics]], '''Wolstenholme's theorem''' states that for a [[prime number]] ''p'' ≥ 5, the congruence
:<math>{2p-1 \choose p-1} \equiv 1 \pmod{p^3}</math>
holds, where the parentheses denote a [[binomial coefficient]].  For example, with ''p'' = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by [[Joseph Wolstenholme]] in 1862.  In 1819, [[Charles Babbage]] showed the same congruence modulo ''p''<sup>2</sup>, which holds for ''p'' ≥ 3. An equivalent formulation is the congruence
:<math>{ap \choose bp} \equiv {a \choose b} \pmod{p^3}</math>
for ''p'' ≥ 5, which is due to [[Wilhelm Ljunggren]]<ref>{{Citation |first=Andrew |last=Granville |title=Binomial coefficients modulo prime powers |journal=Canadian Mathematical Society Conference Proceedings |volume=20 |pages=253–275 |year=1997 |url=http://www.dms.umontreal.ca/%7Eandrew/PDF/BinCoeff.pdf |mr=1483922 |url-status=dead |archive-url=https://web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf |archive-date=2017-02-02}}</ref> (and, in the special case ''b'' = 1, to [[J. W. L. Glaisher]]{{Citation needed|reason=Which paper?|date=May 2020}}) and is inspired by [[Lucas's theorem]].

No known [[composite number]]s satisfy Wolstenholme's theorem and it is conjectured that there are none (see below).  A prime that satisfies the congruence modulo ''p''<sup>4</sup> is called a '''[[Wolstenholme prime]]''' (see below).

As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) [[harmonic number]]s:
:<math>1+{1 \over 2}+{1 \over 3}+\dots+{1 \over p-1} \equiv 0 \pmod{p^2} \mbox{, and}</math>
:<math>1+{1 \over 2^2}+{1 \over 3^2}+\dots+{1 \over (p-1)^2} \equiv 0 \pmod p. </math>

since

:<math>{2p-1 \choose p-1}=\prod_{1\leq k\leq p-1} \frac{2p-k}{k} \equiv 1-2p \sum_{1\leq k\leq p-1} \frac{1}{k} \pmod{p^2}</math>

(Congruences with fractions make sense, provided that the denominators are coprime to the modulus.)
For example, with ''p'' = 7, the first of these says that the numerator of 49/20 is a multiple of 49, while the second says the numerator of 5369/3600 is a multiple of 7.

== Wolstenholme primes ==
{{main|Wolstenholme prime}}

A prime ''p'' is called a Wolstenholme prime [[If and only if|iff]] the following condition holds:

:<math>{{2p-1}\choose{p-1}} \equiv 1 \pmod{p^4}.</math>

If ''p'' is a [[Wolstenholme prime]], then Glaisher's theorem holds modulo ''p''<sup>4</sup>.  The only known Wolstenholme primes so far are 16843 and 2124679 {{OEIS|id=A088164}}; any other Wolstenholme prime must be greater than 10<sup>11</sup>.<ref>{{Cite journal |last=Booker |first=Andrew R. |last2=Hathi |first2=Shehzad |last3=Mossinghoff |first3=Michael J. |last4=Trudgian |first4=Timothy S. | author4-link=Timothy Trudgian| date=2022-07-01 |title=Wolstenholme and Vandiver primes |url=https://link.springer.com/article/10.1007/s11139-021-00438-3 |journal=The Ramanujan Journal |language=en |volume=58 |issue=3 |pages=913–941 |doi=10.1007/s11139-021-00438-3 |issn=1572-9303|arxiv=2101.11157 }}</ref> This result is consistent with the [[heuristic argument]] that the [[Modular arithmetic|residue modulo]] ''p''<sup>4</sup> is a [[pseudo-random]] multiple of ''p''<sup>3</sup>.  This heuristic predicts that the number of Wolstenholme primes between ''K'' and ''N'' is roughly ''ln ln N − ln ln K''.  The Wolstenholme condition has been checked up to 10<sup>11</sup>, and the heuristic says that there should be roughly one Wolstenholme prime between 10<sup>11</sup> and 10<sup>24</sup>.  A similar heuristic predicts that there are no "doubly Wolstenholme" primes, for which the congruence would hold modulo ''p''<sup>5</sup>.

== A proof of the theorem ==

There is more than one way to prove Wolstenholme's theorem.  Here is a proof that directly establishes Glaisher's version using both combinatorics and algebra.

For the moment let ''p'' be any prime, and let ''a'' and ''b'' be any non-negative integers.  Then a set ''A'' with ''ap'' elements can be divided into ''a'' rings of length ''p'', and the rings can be rotated separately. Thus, the ''a''-fold direct sum of the cyclic group of order ''p'' acts on the set ''A'', and by extension it acts on the set of subsets of size ''bp''.  Every orbit of this group action has ''p<sup>k</sup>'' elements, where ''k'' is the number of incomplete rings, i.e., if there are ''k'' rings that only partly intersect a subset ''B'' in the orbit.  There are <math>\textstyle {a \choose b}</math> orbits of size 1 and there are no orbits of size ''p''.<ref>See {{cite web|title=Explanation of the Wolstenholme theorem proof|url=https://math.stackexchange.com/questions/3031711/explanation-of-the-wolstenholme-theorem-proof}} for an explanation.</ref> Thus we first obtain Babbage's theorem
:<math>{ap \choose bp} \equiv {a \choose b} \pmod{p^2}.</math>
Examining the orbits of size ''p''<sup>2</sup>, we also obtain
:<math>{ap \choose bp} \equiv {a \choose b} + {a \choose 2}\left({2p \choose p} - 2\right){a -2 \choose b-1} \pmod{p^3}.</math>
Among other consequences, this equation tells us that the case ''a'' = 2 and ''b'' = 1 implies the general case of the second form of Wolstenholme's theorem.

Switching from combinatorics to algebra, both sides of this congruence are polynomials in ''a'' for each fixed value of ''b''.  The congruence therefore holds when ''a'' is any integer, positive or negative, provided that ''b'' is a fixed positive integer.  In particular, if ''a'' = −1 and ''b'' = 1, the congruence becomes
:<math>{-p \choose p} \equiv {-1 \choose 1} + {-1 \choose 2}\left({2p \choose p} - 2\right) \pmod{p^3}.</math>
This congruence becomes an equation for <math>\textstyle {2p \choose p}</math> using the relation
:<math>{-p \choose p} = \frac{(-1)^p}2{2p \choose p}.</math>
When ''p'' is odd, the relation is
:<math>3{2p \choose p} \equiv 6 \pmod{p^3}.</math>
When ''p'' ≠ 3, we can divide both sides by 3 to complete the argument.

A similar derivation modulo ''p''<sup>4</sup> establishes that
:<math>{ap \choose bp} \equiv {a \choose b} \pmod{p^4}</math>
for all positive ''a'' and ''b'' if and only if it holds when ''a'' = 2 and ''b'' = 1, i.e., if and only if ''p'' is a Wolstenholme prime.

== The converse as a conjecture ==
It is conjectured that if
{{NumBlk|:|<math>{2n-1 \choose n-1} \equiv 1 \pmod{n^k}</math>|{{EquationRef|1}}}}
when ''k'' = 3, then ''n'' is prime.  The conjecture can be understood by considering ''k'' = 1 and 2 as well as 3.  When ''k'' = 1, Babbage's theorem implies that it holds for ''n'' = ''p''<sup>2</sup> for ''p'' an odd prime, while Wolstenholme's theorem implies that it holds for ''n'' = ''p''<sup>3</sup> for ''p'' > 3, and it holds for ''n'' = ''p''<sup>4</sup> if ''p'' is a Wolstenholme prime. When ''k'' = 2, it holds for ''n'' = ''p''<sup>2</sup> if ''p'' is a Wolstenholme prime. These three numbers, 4 = 2<sup>2</sup>, 8 = 2<sup>3</sup>, and 27 = 3<sup>3</sup> are not held for ({{EquationNote|1}}) with ''k'' = 1, but all other prime square and prime cube are held for ({{EquationNote|1}}) with ''k'' = 1. Only 5 other composite values (neither prime square nor prime cube) of ''n'' are known to hold for ({{EquationNote|1}}) with ''k'' = 1, they are called '''Wolstenholme pseudoprimes''', they are

:27173, 2001341, 16024189487, 80478114820849201, 20378551049298456998947681, ... {{OEIS|id=A082180}}

The first three are not prime powers {{OEIS|id=A228562}}, the last two are 16843<sup>4</sup> and 2124679<sup>4</sup>, 16843 and 2124679 are [[Wolstenholme prime]]s {{OEIS|id=A088164}}. Besides, with an exception of 16843<sup>2</sup> and 2124679<sup>2</sup>, no composites are known to hold for  ({{EquationNote|1}}) with ''k'' = 2, much less ''k'' = 3.  Thus the conjecture is considered likely because Wolstenholme's congruence seems over-constrained and artificial for composite numbers.  Moreover, if the congruence does hold for any particular ''n'' other than a prime or prime power, and any particular ''k'', it does not imply that
:<math>{an \choose bn} \equiv {a \choose b} \pmod{n^k}.</math>
The number of Wolstenholme pseudoprimes up to ''x'' is <math>O(x^{1/2} \log(\log(x))^{499712})</math>, so the sum of reciprocals of those numbers converges. The constant 499712 follows from the existence of only three Wolstenholme pseudoprimes up to 10<sup>12</sup>. The number of Wolstenholme pseudoprimes up to 10<sup>12</sup> should be at least 7 if the sum of its reciprocals diverged, and since this is not satisfied because there are only 3 of them in this range, the counting function of these pseudoprimes is at most <math>O(x^{1/2} \log(\log(x))^C)</math> for some efficiently computable constant ''C''; we can take ''C'' as 499712. The constant in the [[big O notation]] is also effectively computable in <math>O(x^{1/2} \log(\log(x))^{499712})</math>.

== Generalizations ==

Leudesdorf has proved that for a positive integer ''n'' coprime to 6, the following congruence holds:<ref>{{cite journal |last=Leudesdorf|first= C. |title=Some results in the elementary theory of numbers |journal=Proc. London Math. Soc. |volume=20 |year=1888 |pages=199–212 |doi=10.1112/plms/s1-20.1.199|url= https://zenodo.org/record/1447726}}</ref>
:<math> \sum_{i=1\atop \gcd(i,n)=1}^{n-1} \frac{1}{i} \equiv 0\pmod{n^2}.</math>

In 1900, Glaisher<ref>J.W.L. Glaisher, Congruences relating to the sums of products of the first n numbers
and to other sums of products, Quart. J. Math. 31 (1900), 1–35.
</ref><ref> J.W.L. Glaisher, On the residues of the sums of products of the first ''p'' − 1 numbers, and
their powers, to modulus p<sup>2</sup> or p<sup>3</sup>, Quart. J. Math. 31 (1900), 321–353.</ref> showed further that: for prime ''p'' > 3, 
:<math> \binom{2p-1}{p-1} \equiv 1- \frac{2p^3}{3}B_{p-3} \pmod{p^4}.</math>

Where ''B<sub>n</sub>'' is the [[Bernoulli number]].

== See also ==
* [[Fermat's little theorem]]
* [[Wilson's theorem]]
* [[Wieferich prime]]
* [[Wilson prime]]
* [[Wall–Sun–Sun prime]]
* [[List of special classes of prime numbers]]
* [[Table of congruences]]

==Notes==
{{reflist}}

==References==
*{{Citation |first=C. |last=Babbage |title=Demonstration of a theorem relating to prime numbers |journal=The Edinburgh Philosophical Journal |volume=1 |year=1819 |pages=46–49 |url=https://books.google.com/books?id=KrA-AAAAYAAJ&pg=PA46}}.
*{{Citation |first=J.W.L. |last=Glaisher |title=Congruences relating to the sums of products of the first ''n'' numbers and to other sums of products |journal=The Quarterly Journal of Pure and Applied Mathematics |volume=31 |year=1900 |pages=1–35 |url=https://books.google.com/books?id=23KWAAAAMAAJ&pg=PA1}}.
*{{Citation |first=J.W.L. |last=Glaisher |title=Residues of binomial-theorem coefficients with respect to p<sup>3</sup> |journal=The Quarterly Journal of Pure and Applied Mathematics |volume=31 |year=1900 |pages=110–124}}.
*{{Citation |first=J.W.L. |last=Glaisher |title=On the residues of the sums of products of the first ''p'' − 1 numbers, and their powers, to modulus p<sup>2</sup> or p<sup>3</sup> |journal=The Quarterly Journal of Pure and Applied Mathematics |volume=31 |year=1900 |pages=321–353}}.
*{{Citation |first=Andrew |last=Granville |title=Binomial coefficients modulo prime powers |journal=Canadian Mathematical Society Conference Proceedings |volume=20 |pages=253–275 |year=1997 |url=http://www.dms.umontreal.ca/%7Eandrew/PDF/BinCoeff.pdf |mr=1483922 |url-status=dead |archive-url=https://web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf |archive-date=2017-02-02}}.
*{{Citation |first=R. J. |last=McIntosh |title=On the converse of Wolstenholme's theorem |journal=Acta Arithmetica |volume=71 |issue=4 |year=1995 |pages=381–389 |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf |doi=10.4064/aa-71-4-381-389|doi-access=free }}.
*R. Mestrovic, [https://arxiv.org/abs/1111.3057 Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862—2012)].
*{{Citation |first=Joseph |last=Wolstenholme |title=On certain properties of prime numbers |journal=The Quarterly Journal of Pure and Applied Mathematics |volume=5 |year=1862 |pages=35–39 |url=https://books.google.com/books?id=vL0KAAAAIAAJ&pg=PA35}}.

== External links ==
* [http://primes.utm.edu/glossary/page.php?sort=Wolstenholme The Prime Glossary: Wolstenholme prime]
* [http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wolstenholme primes]

[[Category:Classes of prime numbers]]
[[Category:Factorial and binomial topics]]
[[Category:Articles containing proofs]]
[[Category:Theorems about prime numbers]]