Talk:Pernicious number

perfect number is Pernicious number

what about 6? I don't see how it fits into formula, which is not clearly stated Dwhjr (talk) 00:18, 19 November 2012 (UTC)

It seems to clear to me when you know what "binary representation" means. For those who don't know it, I have linked it to Binary numeral system here and in the article.[1] The binary representation of 6 is 110₂. This has two 1s, and two is a prime number so 6 is a pernicious number. PrimeHunter (talk) 02:21, 19 November 2012 (UTC)

Properties of Pernicious number

This statement seems valid to me: If a prime p is of the form 2ⁿ − 1, then p is a pernicious number. This is based on the fact that if 2ⁿ − 1 is a prime then n is a prime.Proof

Yes, that was correct. Another user changed it to something incorrect and then I corrected their change without going back to the original version, but that would be OK by me. PrimeHunter (talk) 02:43, 18 September 2012 (UTC)

Your last edit seems fine. I think there should not be any more edits on this property. — Preceding unsigned comment added by Anshuavi (talk • contribs) 03:04, 18 September 2012 (UTC)

2^n + 1

Recently a property was modified[2] with the comment "every number of the form 2ⁿ + 1 is a pernicious number is incorrect; a direct counterexample appears with n = 2)" and then the modified property was deleted[3].

I restored that property, because 2ⁿ + 1 is, in fact, a pernicious number, even in the case of n=2.

What horrible mistake have I made this time? --DavidCary (talk) 16:21, 20 November 2013 (UTC)

The statement is correct, as you now have it. Thank you! The problem is actually when n = 0, for then 2ⁿ + 1 = 2, which has only one one in its binary representation. — Anita5192 (talk) 18:37, 20 November 2013 (UTC)

Good point. To avoid that one special case, the article now says:

"Every number of the form 2n + 1 with n>0, is a pernicious number."

Thank you, everyone. --DavidCary (talk) 03:18, 25 November 2013 (UTC)

Base-dependent numbers?

Why pernicious numbers are base-dependent numbers? They con only be defined in base 2! — Preceding unsigned comment added by 101.12.1.130 (talk) 12:36, 1 September 2015 (UTC)

I have only seen the name used about base 2, but the same idea in base-10 is oeis:A028834: "Numbers n such that sum of digits of n is a prime". If we use the Hamming weight definition and only count the number of non-0 digits then it could also be done in other bases but would be less interesting. PrimeHunter (talk) 01:37, 2 September 2015 (UTC)

Proposed deletion

I've deprodded this (with some trepidation as I'm aware of David Eppstein's qualifications as a mathematician). Possibly this should be merged somewhere, I doubt that there is much research into this, but it has enough coverage to justify being on Wikipedia somewhere. Tianxin Cai in Perfect Numbers And Fibonacci Sequences has "Property 6. Every even perfect number is a pernicious number" followed by a proof and examples [4]. They are also briefly covered by Elena Deza in "Mersenne Numbers And Fermat Numbers" [5]. SpinningSpark 15:15, 17 October 2022 (UTC)

Ok, I think the sources are enough to make this a borderline keep now. It's more or less obvious from the prime number theorem that the number of pernicious numbers up to some threshold $n$ grows like $n/\log \log n$; I wonder if that can be sourced? —David Eppstein (talk) 16:27, 17 October 2022 (UTC)

Nothing to offer on that but I wanted to comment on something else. I strongly suspect that definition in terms of Hamming weight found in Deza is an example of Wikipedia citogenesis. I believe our practice when reliable sources start using terminology through citogenesis is to accept it, even when it actually originated on Wikipedia. SpinningSpark 18:41, 17 October 2022 (UTC)

I think Deza has worked from Wikipedia sourcing before, so I agree that there is cause for suspicion. The "pernicious number" name on OEIS predates Wikipedia, though (you can look up histories on OEIS, and this name was there at the entry's creation in 2000), as does the Colton & Dennis source for the perfect-number property. The classification of 2^n and 2^n+1 were also on the 2000 original OEIS entry. —David Eppstein (talk) 20:04, 17 October 2022 (UTC)

Yes, I wasn't suggesting that pernicious number is citogenesis, only Hamming weight. SpinningSpark 13:11, 18 October 2022 (UTC)

But that's just a standard way of describing the number of nonzeros in a binary sequence. Citogenesis concerns information that cannot be derived any other way than from Wikipedia, not settling on otherwise-standard terminology where it might have been the case that our article chose to use that terminology earlier than other sources. —David Eppstein (talk) 14:32, 18 October 2022 (UTC)