Timeline for Subsets of the integers which are closed under multiplication
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 10, 2021 at 17:33 | answer | added | Acccumulation | timeline score: 1 | |
| Aug 10, 2021 at 17:27 | comment | added | Steven Stadnicki | It's worth noting that 5 isn't quite a special case of 4, and in fact points to a generalization; beyond 'divisible by some set of primes', one can say 'divisible by the set of primes $\{p_i\}$ each with exponent in some additively closed set.' For instance, 5) also includes the numbers which are divisible by primes $\equiv 3\pmod 4$ to even exponent; one could also consider 'divisible by $3^3$' as a requirement. | |
| Aug 10, 2021 at 7:36 | comment | added | Pietro Majer | The set of integers that are not square-free. The set of integers $n=1\mod 13$. | |
| Aug 10, 2021 at 5:41 | comment | added | user7868 | I suspect it's partly because integers have a unique prime factorisation, but there's many ways to write each integer as a sum of other integers. So there's no useful analogue to the OP's example 3 (you can't say "written as a sum of $k$ equal numbers" because that's the same as example 2). | |
| Aug 10, 2021 at 4:46 | history | became hot network question | |||
| Aug 9, 2021 at 23:47 | comment | added | Yaakov Baruch | For any real $k>1$ the set $S_k=\{n\;|\;n\ge \text{rad}(n)^k\}$. | |
| Aug 9, 2021 at 23:18 | comment | added | Anthony Quas | How about this one? let $f$ be any convex increasing function from the reals to itself with the property that $f(0)=0$. Let $S=\{2^a3^b\colon b\le f(a)\}$. | |
| Aug 9, 2021 at 21:28 | answer | added | markvs | timeline score: 15 | |
| Aug 9, 2021 at 21:15 | history | edited | Gautam | CC BY-SA 4.0 |
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| Aug 9, 2021 at 21:01 | comment | added | Noam D. Elkies | The premise is not quite right: the positive integers are also closed under addition, and there are more exotic examples such as all Summer Olympic Games' years (using the official designation of the 32nd games as Tokyo 2020, and assuming no further cancellations beyond 1916, 1940, 1944). But such "additive semigroups" are still more tractable than multiplicative ones because the integers can be generated additively by just $\{1,-1\}$ but require infinitely many multiplicative generators. | |
| Aug 9, 2021 at 20:46 | history | asked | Gautam | CC BY-SA 4.0 |