2013/12/08 by Fred B. Holt, Holt, Fred B., Helgi Rudd +1 · 1 citation
Mathematics · #11A07 #11A41 #11N05 #11N13 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT #msc:11A07 #msc:11A41 #msc:11N05 #msc:11N13
paper · pdf · doi:10.48550/arxiv.1312.2165
arxiv created 2013/12/08 · openalex publication_date 2013/12/08 · arxiv updated 2013/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression. For these initial estimates, we assume that the copies of a given constellation tend toward a uniform distribution in the cycle of gaps, as the recursion progresses. Our simple estimates based on the recursion of gaps and the assumption of uniformity appear to have correct asymptotic behavior, and they exhibit a systematic error correlated to length of the constellation.