The usual product m ^. n on  can be viewed as the sum of n terms  of an arithmetic progression whose first term is a₁ = m − n + 1 and whose difference is d = 2. Generalizing this idea, we define new similar product mappings, and we consider new arithmetics that enable us to extend Furstenberg’s theorem of the infinitude of primes. We also review the classic conjectures in the new arithmetics. Finally, we make important extensions of the main idea. We see that given any integer sequence, the approach generates an arithmetic on integers.

Furstenberg’s proof, arithmetic progression, arithmetic generated by a sequence, polygonal numbers, Peano arithmetic.

Received: September 8, 2021; Accepted: November 10, 2021; Published: November 24, 2021

How to cite this article: F. Javier de Vega, An extension of Furstenberg’s theorem of the infinitude of primes, JP Journal of Algebra, Number Theory and Applications 53(1) (2022), 21-43. DOI: 10.17654/0972555522002

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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