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AN EXTENSION OF FURSTENBERG’S THEOREM OF THE INFINITUDE OF PRIMES
Keywords:
Furstenberg’s proof, arithmetic progression, arithmetic generated by a sequence, polygonal numbers, Peano arithmeticDOI:
https://doi.org/10.17654/0972555522002Abstract
The usual product $m \cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms of an arithmetic progression whose first term is $a_1=m-n+1$ 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.
Received: September 8, 2021
Accepted: November 10, 2021
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