Timeline for The density of the set of non-pathological primes
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 2, 2020 at 11:12 | comment | added | Taras Banakh | @JeppeStigNielsen So, the set of pathological primes seems to be quite dense and there are more pathological primes than non-pathological (in the sense of density)? | |
| Sep 1, 2020 at 10:32 | comment | added | Jeppe Stig Nielsen | Your list of pathological primes up to 157 is incorrect. It should read: $23, 29, 37, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 157, \ldots$. The list of non-pathological primes is identical to OEIS A144755 except for hypothetical Wieferich primes that are non-pathological (both known Wieferich primes are pathological; 1093 goes with 4733, 8861085190774909, and 556338525912325157; and 3511 goes with 1969111, 4242734772486358591 and three larger primes). | |
| Dec 9, 2019 at 21:53 | comment | added | Gerhard Paseman | By the way, $n=11$ is a counterexample to the edit. Bang did show that a primitive prime divisor occurs in 2^n - 1 for $n$ different from 1 and 6, and some of these are not pathological. Gerhard "Some Edits Are Not Pathological" Paseman, 2019.12.09. | |
| Dec 9, 2019 at 21:36 | answer | added | Gerhard Paseman | timeline score: 2 | |
| Dec 7, 2019 at 16:32 | comment | added | Gerhard Paseman | I think you mean 2^n -1 has a primitive prime divisor for each n greater than 6. The likelihood that such a primitive prime is pathologicial is high given large n and the growth of cyclotomic polynomials evaluated at 2 as n grows. (No proof from me, though.) Search MathOverflow for Jameson and cyclotomic to find some relevant notes. I'm guessing Problem 2 is true. Gerhard "And Some Distantly Related Questions" Paseman, 2019.12.07. | |
| Dec 7, 2019 at 11:58 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added info about Bang's Theorem
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| Dec 7, 2019 at 11:55 | comment | added | Taras Banakh | @GerhardPaseman I looked at Zsigmondy Theorem at Wikipedia and found there that for any $n\ne 6$, the number $2^n-1$ has a non-pathological prime divisor. Is this sufficient for stating that the set of non-pathological prime numbers intersects each arithmetic progression $a+bN$ with relatively prime $a,b$? | |
| Dec 7, 2019 at 7:42 | comment | added | Gerhard Paseman | Both. Let's pretend for the moment that there are finitely many Mersenne primes. (If there are infinitely many, I suspect but cannot yet prove they provide a yes answer to 2.) Then I think there are still infinitely many non pathological primes, enough to provide a yes answer to 2, but not enough to be positive density in the primes. Gerhard "Based On Experience, Not Knowledge" Paseman, 2019.12.06. | |
| Dec 7, 2019 at 7:26 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Corrected formulation
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| Dec 7, 2019 at 7:25 | comment | added | Taras Banakh | @GerhardPaseman Yes, I have understood that after posing the problem. But at the moment I am traveling (trains, airports, etc), so not always there is access to the Internet. Of course the density of primes is zero so all subsets of primes also have the density zero. But you suggest that the set of nonpathological primes is small in the set of all primes or large (in the sense of Dirichlet Theorem)? | |
| Dec 7, 2019 at 6:37 | comment | added | Gerhard Paseman | You should talk about relative density, as the Banach density of the primes should itself be zero. You might find results of primitive prime factors (cf. results citing Zsigmondy) useful here. My guess is that (relative to the base set of the primes), the Banach density for non-path. primes is still zero. Gerhard "Also Mersenne Exponent Prime Density" Paseman, 2019.12.06. | |
| Dec 7, 2019 at 6:02 | history | asked | Taras Banakh | CC BY-SA 4.0 |