Timeline for Sets of integers "a little less dense" than the set of prime numbers
Current License: CC BY-SA 4.0
7 events
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| Feb 27 at 7:20 | comment | added | Christian Elsholtz | @StefanKohl I think your approach gives the density you asked for. I agree with Will Sawin that for his suggestion, taking primes with $\lfloor \log p \rfloor$ also prime, it is much easier to work out the details. | |
| Feb 26 at 10:29 | comment | added | Stefan Kohl♦ | @ChristianElsholtz Sorry if I am missing something obvious -- but why does the size of the interval matter here, rather than just the average order of magnitude of the number of prime factors of $p-1$? | |
| Feb 26 at 1:20 | comment | added | Will Sawin | @StefanKohl One could take primes $p$ such that $\lfloor \log p \rfloor$ is also prime. If the log is taken to an integer base this is of course the same as the number of digits, but seems closer to ordinary mathematics. This has the advantage of being easy to rigorously establish - it can be done with just the prime number theorem. If you want to work with the prime factors of $p-1$, a sequence that such work is primes $p$ such that $p-1,p-2,p-3$ all have exactly the same number of prime factors as each other, but this is probably very hard to establish rigorously! | |
| Feb 25 at 22:47 | comment | added | Christian Elsholtz | The typical number of prime factors of $p-1$, where $p$ is about $n$, might be in the interval $[ \log \log n-C \sqrt{\log \log n},\log \log n+C \sqrt{\log \log n} ] $. Hence fixing this number influences the density by a factor of only $\sqrt{\log \log n}$. But you can add the condition that $p \equiv 1 \bmod q$, where $q$ is a prime. Then $\sum_{p<x} \frac{1}{p} \sim \frac{\log \log x}{q-1}$. Choosing $q\sim \frac{\log \log x}{\log \log \log x}$ works. (This can be arranged, by upper bounds on prime gaps.) | |
| Feb 25 at 21:46 | comment | added | Stefan Kohl♦ | To avoid looking at digit counts -- wouldn't the set of primes $p$ which are congruent to 1 modulo the number of prime factors of $p-1$ work as well? | |
| Feb 25 at 21:29 | history | edited | Christian Elsholtz | CC BY-SA 4.0 |
added 12 characters in body
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| Feb 25 at 21:20 | history | answered | Christian Elsholtz | CC BY-SA 4.0 |