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Jon Awbrey (Aug 28 2021 at 14:36):Cf : Riffs and Rotes • 1
Re: Richard J. Lipton • Making Primes More Random
There’s a study called generalized primes which investigates in a more general way the relationship between arbitrary elements called primes and the composites which can be formed from them according to specified rules of composition. Comparisons can be made among a variety of numerical systems or any orders of combinatorial species one might imagine. I seem to recall at least one old monograph by Rademacher on the subject.
My fascination with questions like that led me many years ago to the Riff and Rote trick, a special case of the Make A Picture trick. There's a bit on that in the following article.
Jon Awbrey (Aug 29 2021 at 18:08):Re: Peter Cameron • Addition and Multiplication of Natural Numbers
The interaction between addition and multiplication in the natural numbers has long been an interest of mine, leading to broader questions about the relationship between algebra and combinatorics. My gropings with those enigmas led me to the structures of Riffs and Rotes, extracting what we might think of as the “purely combinatorial” properties of primes factorizations. Thinking of the additive structure of the positive integers as embodied in their total linear ordering, the following two questions arise.
Reference