Talk:Cullen number

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Field: Number theory

weak property

It migt be true, that a Cullen number Cp is divisible by 2p-1, if p is a prime and ${\displaystyle p=8k-3}$. But there exist other Cp is divisible by 2p-1, where p is not equal to 8k-3. At last there exist n there neither prime nor of the form 8k-1, but Cn is divisble by 2n-1.

``` n   c(n)  2n-1  8k-3   d/n
----------------------------
1      3     1           d
2      9     3           d
3     25     5     1     d
4     65     7           n
5    161     9           n
6    385    11           d
7    897    13     2     d
8   2049    15           n
9   4609    17           n
10  10241    19           d
11  22529    21           n
12  49153    23           d
```

I think, this is a very weak property. --Arbol01 18:07, 11 Feb 2005 (UTC)

almost all

At first reading, the second paragraph sounded self-contradictory to me. But I saw that the page almost all indicates alternate meanings besides the "all but finitely many" that I'm used to in number theory. I would recommend specifying which definition of "almost all" you are referring to, such as almost all Cullen numbers are composite, in the sense that the number of Cullen numbers less than x, divided by x, approaches zero as x approaches infinity.

Johnny Vogler (talk) 21:30, 26 October 2008 (UTC)

I found a freely available copy of the full article by Wilfrid Keller, "New Cullen Primes" at ams.org (the one originally linked to required subscription). It explains the basics of the proof by C. Hooley and mentions an extension by H. Suyama. I updated the article accordingly, and changed the reference to the new URL. Adcoon (talk) 07:50, 5 September 2010 (UTC)

Dual Cullen number

Let n<0, the numerator of the number n*2n + 1 is 2n - n, the first few of them are

1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556, 2097131, 4194282, 8388585, 16777192, ... (sequence A000325 in the OEIS)

Numbers n such that 2n - n is prime are

2, 3, 9, 13, 19, 21, 55, 261, 3415, 4185, 7353, ... (sequence A048744 in the OEIS)

The dual Woodall prime themselves are

2, 5, 503, 8179, 524269, 2097131, ... (sequence A081296 in the OEIS)

I have removed a section called "Dual Cullen number" with the above content. It was addded today by 49.215.193.130. Reasons for removal:

1. Google gives no hits on "Dual Cullen number" or "dual Woodall prime"
2. After apparently inventing the name "Dual Cullen number", I don't see reason to say "dual Woodall prime" about such numbers which are prime.
3. I cannot find sense in "the numerator of the number n*2n + 1 is 2n - n". For example, for n = -6 it becomes "the numerator of the number 29/32 is 385/64".
4. The OEIS sequences don't mention a connection to Cullen numbers.

PrimeHunter (talk) 23:44, 29 May 2015 (UTC)