History of Patterns among the Primes
Fred B. Holt
& Helgi Rudd
Fred began studying the cycles of gaps G(p#) in 1994. This project went in-and-out of a desk drawer for several years. From small cycles, like G(11#) or G(13#), he was able to outline preliminary results and first posted these to arXiv in 2007. Helgi Rudd saw this paper and jumped in to extend the computations, to experiment with different visualizations, and to brainstorm about the patterns they were seeing. Helgi generated the cycles of gaps up to G(37#) and produced counts of the populations of gaps and constellations in the cycles G(p#) and between primes over this range.
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One of the most important contributions that Helgi made was to ignore the upper bounds that Fred requested for certain computations. Why stop at N when we can get the data for 10N ? Helgi barreled through constraints that didn't really exist, freeing our imagination to expand our studies.
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They collaborated on this project from late 2007 into 2015, when Helgi suddenly passed away. Their paper Combinatorics of the gaps between primes summarizing their work-to-date would be presented at the Connections in Discrete Mathematics conference at Simon Frasier University in June 2015.
Dedication to Helgi Rudd
These studies benefitted greatly from Helgi's enthusiasm, curiosity, and computer skills. He helped propel these studies forward. The computational and theoretical aspects fed off each other, and together we were able to explore several questions about the patterns across these cycles G(p#).
Book
Videos
Exercises
Papers
Useful Resources
Contact
All admissible k-tuples arise and persist in Eratosthenes sieve, presentation slides from JMM2025, Jan 2025.
Patterns among the Primes: studying Eratosthenes sieve as a discrete dynamic system, presentation slides from JMM2024, Jan 2024.
Expected biases in the distribution of consecutive primes, arXiv:2405.03540v1, Apr 2024.
On the counts of p-rough numbers, arXiv:2308.07570, Nov 2023.
Models for gaps g=2p1, arXiv:2309.16833, Oct 2023.
Combinatorics of the gaps between primes, arXiv:1510.00743, presented at Connections in Discrete Mathematics, June 2015.
The combinatorics of gaps between prime numbers, presentation slides at Connections in Discrete Mathematics, June 2015.
Eratosthenes sieve and the gaps between primes, arXiv:1408.6002, Aug 2014.
On the last digits of consecutive primes, arXiv:1604.02443, Apr 2016.
On Polignac's Conjecture, arXiv:1402.1970, Feb 2014.
On small gaps among primes, arXiv:1312.7569, Dec 2013.
Estimating constellations among primes - I.Uniformity, arXiv:1312.2165, Dec 2013.
Expected gaps between prime numbers, arXiv:0706.0889, Jun 2007.
Thomas Nicely's prime web pages.
Chris Caldwell's prime pages
Thomas Engelsma on narrow or dense k-tuples
Email - discuss@primegaps.info