1994/01/01 by David W. Boyd · 46 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics #Conjecture #Discrete mathematics #Mathematical Dynamics and Fractals #Mathematics #Natural density #Prime (order theory) #Prime number #Series (stratigraphy)
paper · pdf · doi:10.1080/10586458.1994.10504298
published in Experimental Mathematics 3(4), 287-302 (Taylor & Francis)
openalex publication_date 1994/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Let H n = 1 + ½ + … + be the n-th partial sum of the harmonic series. A classical result of Wolstenholme states that, if p > 3 is prime, the numerator of H p –l is divisible by p 2. Here we consider, for a given prime p, the set J p of n for which p divides the numerator of H n . This set J p had been previously determined for p = 2,3,5,7. One of our results is that J 11 contains exactly 638 integers, the largestof which is a number of 31 decimal digits. We determine J p for all p < 550 with three exceptions: 83, 127 and 397. The computation is based on a new p-adically convergent formula for the quantity H pn – H n /p. We describe a probabilistic model for the sets J p , based on branching processes. The model predicts that |J p | = O(p 2(log log p)2+∊), and that there are infinitely many p with |J p | ≥ p 2(log log p)2. This strengthens an earlier conjecture of Eswarathasan and Levine that |J p| is finite for all p. Another prediction of the model is that there will be infin itely many pairs (n,p) for which p 3 divides the numerator of H n , but only finitely many for which p 4 divides H n . It has been conjectured that there are infinitely many p for which |Jp | = 3. We give a probabilistic argument that suggests that such primes have a density 1/e in the set of all primes, and experimentally confirm this by a determination of all such p ≤ 105.