Timeline for Is there some example that nicely extends the multiplication of natural numbers?

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Feb 17, 2022 at 0:17	comment	added	Zelox		And your argument of the non-existence of non-unit common divisor $x$ only holds when $S$ is a ring on which subtraction is allowed. But for semiring like $\mathbb N$, the equation $ap+bq = 1$ has no solution.
Feb 17, 2022 at 0:14	comment	added	Zelox		Thanks, your mentioning of $p$-adic integers reminds me that I forget to require the factorization of primes to be non-trivial. By doing this, $f(p)$ is non-unit for any $p \in \mathbb P$. Your remaining confusions all come from the "simpler" condition. One, but not the only, way to interpret it is to think there is a neat formula for the primes. So I added an alternative condition (3'). I hope it is clear now.
Feb 16, 2022 at 13:44	comment	added	Yaakov Baruch		I don't fully understand what the OP is looking for either. But let's say in $\mathbb{P}$ we replaced some primes with one of their roots, in such a way that the resulting prime gaps are still always $\ge 1$, but never, say, greater than $2\log(p)$ (to the left or right of $p$)... and that that could be done with a neat (rather than ad hoc) algorithm for picking the roots... would that perhaps be a "simpler " set of primes?
Feb 16, 2022 at 13:09	history	answered	Timothy Chow	CC BY-SA 4.0