Wolstenholme and Vandiver primes

In 1862, Wolstenholme [Wolstenholme] established that the following three congruences hold for every prime $p\geq 5$:

These results are closely related: if the stronger congruence obtained by replacing the modulus $p^{m}$ by $p^{m+1}$ holds in one of the expressions above for a particular prime $p$, then a similar strengthening for that same prime $p$ holds in all of them. See [Gardiner] for a proof, [AebiCairns] for remarks on higher-order congruences, and [MestrovicSurvey] for a survey on other generalizations and extensions of Wolstenholme’s theorem. A Wolstenholme prime (first defined by McIntosh [McIntosh]) is an odd prime $p$ for which any of the following congruences holds:

Many conditions equivalent to these are also well known. For example, in the last congruence, the constant $2$ may be replaced by any integer $h$ not divisible by $p$. It is also straightforward to rewrite that generalized congruence as

which some authors employ.

Recall the Bernoulli numbers $B_{k}$ may be defined by their exponential generating function,

These are rational numbers, and $B_{2k+1}=0$ for $k\geq 1$. An important connection between certain binomial coefficients and Bernoulli numbers was established by Glaisher in 1900 (combining [Glaisher1, §39] and [Glaisher2, §8]), who proved that

for any prime $p\geq 5$ and any positive integer $h$. Thus, $p$ is a Wolstenholme prime if and only if $p$ divides the numerator of the Bernoulli number $B_{p-3}$. This condition provides the first observation toward a more computationally attractive test for Wolstenholme primes, since it requires computation of a residue mod $p$, rather than a higher power of $p$.

Glaisher’s observation implies that a Wolstenholme prime is a particular type of irregular prime. Recall that a prime $p$ is said to be regular if $p$ does not divide the class number of the cyclotomic field $\mathbb{Q}(\zeta_{p})$, where $\zeta_{p}=e^{2\pi i/p}$. Equivalently, $p$ is regular if $p$ does not divide the numerator of any of the Bernoulli numbers $B_{2k}$ with $0<2k\leq p-3$. Regular primes were first defined in the context of work on Fermat’s Last Theorem, as Kummer proved in 1847 that no nontrivial solutions with exponent $p$ exist in this famous problem if $p$ is a regular prime. The smallest irregular prime is $37$, and while it is known that there exist infinitely many irregular primes (see for instance \citelist[Carlitz54][LPMP15]), it remains unknown if the number of regular primes is infinite. We refer the reader to [Edwards, Washington] for more information on regular primes and their connections to various topics in number theory.

The Euler numbers $E_{k}$ may also be defined by their exponential generating function,¹¹1Some authors instead use the series for the hyperbolic secant, $\operatorname{sech}z=\frac{2e^{z}}{e^{2z}+1}$, to define the Euler numbers; this alters the sign of the terms with index congruent to $2$ mod $4$. Results cited from the literature that employ that formulation are suitably translated here to comport with the definition we employ.

Like the Bernoulli numbers, the Euler numbers with odd index are $0$ (including $E_{1}$ this time), but unlike the $B_{k}$, the Euler numbers are all integers. A prime $p$ is said to be $E$-regular if $p$ does not divide any of the Euler numbers $E_{2k}$ with $0<2k\leq p-3$. The smallest $E$-irregular prime is $19$, as $19\mid E_{10}$, and while it is known that there exist infinitely many $E$-irregular primes \citelist[Carlitz54][LPMP15], it is also not known if the number of $E$-regular primes is infinite. $E$-irregular primes first arose in the study of Fermat’s Last Theorem as well, and they are pertinent in other problems in number theory too, including the topic of class numbers of cyclotomic fields. For example, Gut [Gut] proved that $p$ does not divide the class number of the cyclotomic field $\mathbb{Q}(\zeta_{4p})$ if and only if $B_{2k}\not\equiv 0$ and $E_{2k}\not\equiv 0$ mod $p$ for $0<2k\leq p-3$.

The extremal case $E_{p-3}\equiv 0$ mod $p$ has witnessed particular interest. In 1940, Vandiver [Vandiver40] showed that any solution with exponent $p$ in the first case of Fermat’s Last Theorem (so a nontrivial solution in positive integers to $x^{p}+y^{p}=z^{p}$ where $p\nmid xyz$) necessarily has $p\mid E_{p-3}$. More recently, Z.-H. Sun [ZHSun08] and Z.-W. Sun [ZWSun11] established a number of congruences regarding Bernoulli and Euler numbers, in many of which the residues of $B_{p-3}$ and $E_{p-3}$ mod $p$ play an important role, and several of these congruences become stronger in the case of a Wolstenholme or Vandiver prime.

If $p$ is irregular, and $B_{2r}\equiv 0$ mod $p$ for an even integer $2r\leq p-3$, then $(p,2r)$ is known as an irregular pair. A Wolstenholme prime is thus the extremal case where $(p,p-3)$ forms an irregular pair. Only two Wolstenholme primes are known, and both were discovered during broader searches for irregular primes. The smallest Wolstenholme prime is $p_{1}=16843$. This seems to have been first discovered by Selfridge and Pollack, who spoke on their work searching for irregular primes up to $25000$ at a meeting of the American Mathematical Society in 1964 [SelfridgePollack]. While details from their search are not readily available, in 1975 Johnson [Johnson75] reported on an extension of the work of Selfridge and Pollack, verifying their work, and noted in particular that $p_{1}$ was the only prime less than $30000$ where $(p,p-3)$ forms an irregular pair.

The second Wolstenholme prime is $p_{2}=2124679$, and was discovered in 1993 by Buhler et al. [BCEM93] during a search for irregular primes $p<4\cdot 10^{6}$. In 1995 McIntosh [McIntosh] reported finding no additional examples with $p<2\cdot 10^{8}$, after more extensive searches that targeted Wolstenholme primes in particular, and then (with Roettger) extended this search in 2007 [McIntoshRoettger] to $p<10^{9}$. More recently, McIntosh continued this search to cover $p<10^{10}$ [McIntoshFQ] and found no additional examples. Broader searches for irregular primes have continued as well, culminating most recently in the work of Hart, Harvey, and Ong [HHO] of 2017, who determined all irregular primes $p<2^{31}$, along with their index of irregularity, that is, the number of irregular pairs attached to each such prime. They found for instance that the maximum index of irregularity over this range is $9$, attained by the single prime $p=1767218027$.

We likewise say $(p,2r)$ is an $E$-irregular pair if $E_{2r}\equiv 0$ mod $p$, so that a Vandiver prime has the property that $(p,p-3)$ forms an $E$-irregular pair. It appears that the last census of $E$-irregular primes and indices of $E$-irregularity was performed in 1978 by Ernvall and Metsänkylä \citelist[Ernvall79][EM78], who checked primes $p<10^{4}$, motivated by questions regarding Iwasawa invariants of the cyclotomic field $\mathbb{Q}(\zeta_{4p})$. They found for example that the maximum index of $E$-irregularity over this range was $5$, attained by $p=5783$ alone.

A number of Vandiver primes were known prior to this work. The first two, $p=149$ and $241$, were first noted by Ernvall and Metsänkylä [EM78] (and later rediscovered by Z.-W. Sun [ZWSun11]). Three more Vandiver primes, $p=2124679$, $16467631$, and $17613227$, were found by Cosgrave and Dilcher in 2013 [CosgraveDilcher] (who called these integers $E$-primes), using a congruence from [ELehmer] to search up to $5\cdot 10^{7}$. They showed that these primes arise in a natural way in a problem regarding modular sums of certain reciprocals, proving that

precisely when $n=45$ or $n$ is divisible by a Vandiver prime. Cosgrave and Dilcher also reported that in 2012 McIntosh found three additional such primes in a search to $3\cdot 10^{9}$: $p=327784727$, $426369739$, and $1062232319$. More recently, McIntosh extended this to $5\cdot 10^{9}$ [McIntoshFQ], finding no additional examples. We also report that in 2014 Meštrović [Mestrovic14] employed a congruence of Z.-H. Sun from [ZHSun08] involving the computation of certain harmonic numbers modulo $p^{2}$ to compute the Vandiver primes $p<10^{7}$.

In the absence of any special structure, one might expect the values of $B_{p-3}$ or $E_{p-3}$ to be uniformly distributed modulo $p$.²²2We remark that some structure does appear in a similar problem. The Ankeny–Artin–Chowla conjecture asserts that $p\nmid B_{(p-1)/2}$ for every prime $p\equiv 1$ mod $4$. While this problem is open in general (and has been verified computationally for $p<2\cdot 10^{11}$ [VTW]), this conjecture is known to hold for primes of the form $p=n^{2}+1$ or $n^{2}+4$ [Agoh]. Numerical evidence weighs in support of this: see Figures 1 and 2 later in this paper. If this is true, then as McIntosh [McIntosh] has pointed out, one would expect that infinitely many of each of these kinds of primes exist, since the sum of the reciprocals of the primes diverges. In this case, the expected number of either type of prime up to $x$ would grow approximately as $\log\log x$. One might define a super-Wolstenholme prime by requiring a congruence in (1.1) to hold modulo a higher power of $p$, and similarly for a super-Vandiver prime. Similar heuristic considerations lead one to expect that only a finite number of such primes exist. In [McIntosh] McIntosh conjectured that no super-Wolstenholme primes exist; this seems likely to be true as well for super-Vandiver primes.

In this article, we describe searches for Wolstenholme and Vandiver primes that reach eight times farther than any prior search in the former case, and twelve times farther in the latter. We establish that no additional examples of either type occur across the ranges we check.

There are two principal components to our method which allowed us to extend these searches by substantial factors. First, we obtain new congruences for Bernoulli and Euler numbers modulo a prime $p$, which allow us to compute residues of these numbers with greater efficiency. See (2.8), (2.9), (2.10), (2.11), and (2.12) for successively more efficient (and more complicated) congruences for the Bernoulli numbers, and similarly see (4.2), (4.3), (4.4), (4.5), (4.6), and (4.9) for our congruences for the Euler numbers. For the Bernoulli numbers, our congruences extend and improve on some results of Tanner and Wagstaff from 1987 [TannerWagstaff]. Second, we employ graphics processing units (GPUs) to implement a heavily parallelized approach.

We also answer in the affirmative a question raised by Tanner and Wagstaff regarding the existence of congruences for Bernoulli numbers mod $p$ involving fewer than $\epsilon p$ terms, for arbitrarily small $\epsilon>0$—see Theorem 6.1. This result applies equally well to the computation of Euler numbers mod $p$.

This article is organized in the following way. Section 2 derives a number of new congruences for Bernoulli numbers which are advantageous for computing, particularly for large primes. Section 3 then describes our search for Wolstenholme primes using these congruences and summarizes the results of our searches in this problem up to $10^{11}$. Section 4 establishes some new congruences amenable for our computations on Euler numbers, and Section 5 describes our search for Vandiver primes and summarizes our results up to this same bound. Section 6 establishes that congruences with fewer than $\epsilon p$ terms exist for computing residues of Bernoulli or Euler numbers, for arbitrary $\epsilon>0$, and Section 7 adds some observations regarding derivations of favorable congruences.

The Bernoulli numbers are known to satisfy a number of congruences involving sums of powers of consecutive integers, beginning with Voronoi’s well-known congruence of 1889 (see [UspenskyHeaslet, Chap. IX]). Some of these congruences have been particularly instrumental in calculations involving Bernoulli numbers. We review some of the more important relations. For real numbers $x<y$ in $[0,1]$, an integer $\ell$, and a prime $p$, let

and, for positive integers $a$, $b$, $c$ and a positive integer $k$, let

With this notation, in 1930 Stafford and Vandiver [StaffordVandiver, eq. (9)]] (see also [TannerWagstaff, Thm. 1]) established that

whenever $p\geq 5$ and $p-1\nmid 2k$. Note that this allows computing the residue of $B_{2k}$ mod $p$ using a sum with approximately $p/12$ terms. Also, in 1937, Vandiver [Vandiver37, eq. (18)] proved that

when $p\geq 7$ and $p-1\nmid 2k$. For convenience, we define the cost of a congruence such as (2.2) or (2.3), where in general the right side consists of a linear combination of sums $S_{\ell}(x_{1},y_{1})$, …, $S_{\ell}(x_{m},y_{m})$, as

as this is the approximate number of terms in the sums comprising the right side of the congruence. The cost of Vandiver’s congruence (2.3) is thus $p(\frac{1}{5}-\frac{1}{6}+\frac{2}{5}-\frac{1}{3})=p/10$, and the cost of (2.2) is $p/12$. However, it transpires that (2.3) is more important for our application.

In 1987, Tanner and Wagstaff [TannerWagstaff] developed a number of new congruences for Bernoulli numbers. They proved for example that

whenever $p$ is prime, $b\geq 2$, $p>b+1$, $k\geq 1$, and $p-1\nmid 2k$. This rule has cost $\frac{\left\lfloor b/2\right\rfloor(\left\lfloor b/2\right\rfloor+1)}{2b(b+1)}p$, so asymptotically $p/8$, and we note that the case $b=5$ recovers (2.3). Tanner and Wagstaff also noted three simple transformations that preserve the value of $S_{\ell}(x,y)$ mod $p$, which we summarize here, in a slightly generalized form.

We include the short proof here for the convenience of the reader.

One may employ the transformations of Proposition 2.1 to create new congruences for Bernoulli numbers from existing ones. While each transformation preserves the cost of the sums involved, one might hope that subdivision and reflection might create new expressions with some overlapping sums, so that separation may be employed to collect like terms, and thus produce congruences with reduced cost, at the price of some more complicated coefficients. For example, as noted in [TannerWagstaff], if one applies subdivision with $d=2$ to the second term of (2.3), and then reflection to the resulting term having $x>1/2$, then one obtains

provided $p\geq 7$ and $p-1\nmid 2k$. This new expression has two sums, each now possessing modestly more complicated coefficients, but the cost of this congruence has dropped to $p/15$. Tanner and Wagstaff also derived a more complicated congruence for Bernoulli numbers with more sums and lower cost. By starting with (2.4) in the case $b=9$, and employing five prescribed subdivision steps in sequence, using reflection and separation as appropriate for simplification, they found a congruence for $B_{2k}$ mod $p$ having seven sums with cost $p/18$. They also reported that beginning with the congruence (2.5) and applying two particular subdivision operations, together with appropriate simplifications, produces an expression with cost $p/19.2$. While the resulting congruence is not reported, we find that it has eight sums. Finally, it is mentioned in the same article that the authors could construct a more complicated congruence with approximate cost $p/22$, but its construction is not specified.

We use Proposition 2.1 to search for congruences for Bernoulli numbers mod $p$ that are particularly favorable for computations. Certainly, it is advantageous to employ a congruence with small cost, but one must also be concerned with the number of sums appearing in the expressions, as well as the size and complexity of their coefficients, as each of these contributes to overhead costs in computations. It may also be helpful for each sum in a congruence to have a comparable number of elements, for balancing loads in parallel environments. In addition, for larger primes we may be able to tolerate greater overhead costs, so we therefore seek a family of congruences for Bernoulli numbers mod $p$: for a range of small positive integers $m$, we would like an expression for $B_{2k}$ having exactly $m$ sums with cost $p/r_{m}$, with $r_{m}$ as large as possible.

The case $m=1$ was the subject of a search by Wagstaff [Wagstaff], who found nothing better than the value $r_{1}=12$ exhibited by (2.2), as well as by the well-known relation (see, e.g., [UspenskyHeaslet, ex. IX.6])

valid for primes $p\geq 5$ when $p-1\nmid 2k$. This latter relation follows from Voronoi’s congruence, and in fact is the case $b=3$ in (2.4). We searched for improved values for some larger $m$ by using Proposition 2.1, with a number of starting relations.

We employed two strategies in searching for useful congruences: exhaustive searches, and heuristic searches. We describe each of these below, along with its principal results.