[Paper Review] Congruences for Wolstenholme primes

Romeo Meštrović|arXiv (Cornell University)|Aug 21, 2011

Analytic Number Theory Research8 references4 citations

TL;DR

This paper establishes higher-order congruences for binomial coefficients ${2p-1 inom{p-1}}$ modulo $p^8$ and $p^7$ for Wolstenholme primes, expressing them in terms of harmonic sums $R_i = \sum_{k=1}^{p-1} \frac{1}{k^i}$ and Bernoulli numbers. The key contribution is a new characterization of Wolstenholme primes via the congruence ${2p-1 \choose p-1} \equiv 1 - 2p\sum_{k=1}^{p-1}\frac{1}{k} - 2p^2\sum_{k=1}^{p-1}\frac{1}{k^2} \pmod{p^7}$, which implies the prime must be a Wolstenholme prime.

ABSTRACT

A prime number $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\choose p-1} \equiv 1 \,\,(\bmod{\,\,p^4})$. For such a prime $p$, we establish the expression for ${2p-1\choose p-1}\,\,(\bmod{\,\,p^8})$ given in terms of the sums $R_i:=\sum_{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime, $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} -2p^2\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^7}. $$ Moreover, using a recent result of the author \cite{Me}, we prove that the above congruence implies that a prime $p$ necessarily must be a Wolstenholme prime. Applying a technique of Helou and Terjanian \cite{HT}, the above congruence is given as the expression involving the Bernoulli numbers.

Motivation & Objective

• To derive higher-order congruences for the binomial coefficient ${2p-1 \choose p-1}$ modulo $p^8$ for Wolstenholme primes.
• To express these congruences in terms of harmonic sums $R_i = \sum_{k=1}^{p-1} \frac{1}{k^i}$ for $i=1,2,3,4,5,6$.
• To reduce the expression to involve only $R_1, R_3, R_4, R_5$ and further to $R_1, R_2, R_3, R_5$ for simpler forms.
• To establish a new characterization of Wolstenholme primes via a congruence modulo $p^7$ that implies the prime must satisfy the Wolstenholme condition.
• To connect the congruences to Bernoulli numbers using Kummer congruences and known results from prior work.

Proposed method

• Derive a general expansion of ${2p-1 \choose p-1}$ modulo $p^8$ using the identity involving harmonic sums $R_i$ up to $i=6$.
• Apply the assumption that $p$ is a Wolstenholme prime, i.e., ${2p-1 \choose p-1} \equiv 1 \pmod{p^4}$, to simplify the expansion.
• Use known congruences for harmonic sums modulo $p^k$ and the fact that $p$ divides the numerator of $B_{p-3}$ to reduce terms.
• Employ Kummer congruences to relate Bernoulli numbers $B_{p^n - p^{n-1} - s}$ to lower-order Bernoulli numbers modulo $p^n$.
• Substitute series expansions of $1/(kp - m)$ modulo $p^4$ to handle rational coefficients in the Bernoulli number expressions.
• Combine results from prior work, including a recent theorem by the author [Me, Theorem 1.1], to establish a converse characterization of Wolstenholme primes.

Experimental results

Research questions

• RQ1 Can the binomial coefficient ${2p-1 \choose p-1}$ be expressed in a refined form modulo $p^8$ for Wolstenholme primes?
• RQ2 Does the congruence ${2p-1 \choose p-1} \equiv 1 - 2p\sum_{k=1}^{p-1}\frac{1}{k} - 2p^2\sum_{k=1}^{p-1}\frac{1}{k^2} \pmod{p^7}$ uniquely characterize Wolstenholme primes?
• RQ3 Can the expression for ${2p-1 \choose p-1}$ modulo $p^7$ be rewritten using only selected harmonic sums $R_1, R_3, R_4, R_5$?
• RQ4 How can the congruence be reformulated in terms of Bernoulli numbers, and what role do Kummer congruences play in this?
• RQ5 Is there a deeper structural link between the vanishing of certain harmonic sums and the Wolstenholme property?

Key findings

• The paper derives a full expression for ${2p-1 \choose p-1} \pmod{p^8}$ for Wolstenholme primes in terms of harmonic sums $R_1$ through $R_6$, showing explicit dependence on $p^i R_i$ for $i=1$ to $6$.
• The expression is simplified to depend only on $R_1, R_3, R_4, R_5$, yielding the congruence ${2p-1 \choose p-1} \equiv 1 + \frac{3p}{2}R_1 - \frac{p^2}{4}R_2 + \frac{7p^3}{12}R_3 + \frac{5p^5}{12}R_5 \pmod{p^8}$.
• Reducing the modulus to $p^7$, the paper proves ${2p-1 \choose p-1} \equiv 1 - 2pR_1 - 2p^2R_2 \equiv 1 + 2pR_1 + \frac{2p^3}{3}R_3 \pmod{p^7}$.
• The congruence modulo $p^7$ is re-expressed using Bernoulli numbers as ${2p-1 \choose p-1} \equiv 1 - p^3 B_{p^4 - p^3 - 2} - \frac{3}{2}p^5 B_{p^2 - p - 4} + \frac{3}{10}p^6 B_{p-5} \pmod{p^7}$.
• A further expansion expresses the same congruence in terms of lower-order Bernoulli numbers: $-p^3(\frac{8}{3}B_{p-3} - 3B_{2p-4} + \frac{8}{5}B_{3p-5} - \frac{1}{3}B_{4p-6}) - p^4(\frac{8}{9}B_{p-3} - \frac{3}{2}B_{2p-4} + \frac{24}{25}B_{3p-5} - \frac{2}{9}B_{4p-6}) - p^5(\frac{8}{27}B_{p-3} - \frac{3}{4}B_{2p-4} + \frac{72}{125}B_{3p-5} - \frac{4}{27}B_{4p-6} + \frac{12}{5}B_{p-5} - B_{2p-6}) - \frac{2}{25}p^6 B_{p-5} \pmod{p^7}$.
• The paper proves that the congruence ${2p-1 \choose p-1} \equiv 1 - 2pR_1 - 2p^2R_2 \pmod{p^7}$ implies that $p$ must be a Wolstenholme prime, providing a new necessary and sufficient condition for this rare class of primes.