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Primeinator 2009-07-18 02:52

Riesel primes
 
For Riesel primes of the form k*2^n -1, I know K is an odd integer. What are the restrictions for the value of n?

Mini-Geek 2009-07-18 02:54

I'm pretty sure it's only that n is such that 2^n>k (otherwise every number would be a Riesel number, and every prime a Riesel prime).

Primeinator 2009-07-18 05:54

[QUOTE=Mini-Geek;181486]I'm pretty sure it's only that n is such that 2^n>k (otherwise every number would be a Riesel number, and every prime a Riesel prime).[/QUOTE]

How would 2^n < k in of itself make a number riesel? I thought that any number of the form k*2^n -1 is a riesel number.

Mini-Geek 2009-07-18 12:34

[quote=Primeinator;181504]How would 2^n < k in of itself make a number riesel? I thought that any number of the form k*2^n -1 is a riesel number.[/quote]
I was mistaken about every number being represented by some k*2^n-1. Every [I]odd[/I] number can be represented by k*2^n-1 if 2^n can be less than k. Let's choose 15648132147819545 for example.
15648132147819545=k*2^n-1
15648132147819546=k*2^n
15648132147819546 has only one factor of 2, so we set n to 1
2*7824066073909773=k*2
7824066073909773=k
So 15648132147819545=7824066073909773*2^1-1

In case you're wondering, there is an identical requirement for Proth (k*2^n[B]+[/B]1) numbers.
[url]http://en.wikipedia.org/wiki/Proth_number[/url]

Primeinator 2009-07-18 15:23

[QUOTE=Mini-Geek;181542]I was mistaken about every number being represented by some k*2^n-1. Every [I]odd[/I] number can be represented by k*2^n-1 if 2^n can be less than k.

In case you're wondering, there is an identical requirement for Proth (k*2^n[B]+[/B]1) numbers.
[url]http://en.wikipedia.org/wiki/Proth_number[/url][/QUOTE]

Now that makes sense. And of course, with the exception of two, being odd is a necessity for any number x being a Riesel prime.

However, you can still get an odd number by having n < k. A simple example:

3*2^2 -1 = 11 = ! prime.
15*2^3 - 1 = 119 = ! prime

Any number of k*2^n -1 will be odd, given k is odd. For n > k a prime can turn up. The first is k =3 and n = 6 which yields the following
3*2^6 - 1 = 191 = ! prime.

Finally, n = k is also a possibility. 3*2^3 -1 = 23 = ! prime.

Apparently, all three scenarios are possible?

kar_bon 2009-07-18 15:50

if you search for a comprehensive collection of Riesel primes have a look at
[url]www.rieselprime.de[/url]

BTW: n=k are called Woodall primes (see above link also)

Primeinator 2009-07-18 16:05

[QUOTE=kar_bon;181570]if you search for a comprehensive collection of Riesel primes have a look at
[url]www.rieselprime.de[/url]

BTW: n=k are called Woodall primes (see above link also)[/QUOTE]

From this site it appears that N is always bigger than k. However, even though the opposite can still produce a prime, is it considered a Riesel prime?

kar_bon 2009-07-18 16:19

[QUOTE=Primeinator;181573]From this site it appears that N is always bigger than k. However, even though the opposite can still produce a prime, is it considered a Riesel prime?[/QUOTE]

mhhh?

have a look at the range for example 2000<k<4000 in the Data Section.
only k=2001 got 24 primes with n<k!!!

Mini-Geek 2009-07-18 16:49

Remember that the restriction is 2^n>k, not n>k, so e.g. n=4 k=15 is allowed (2^4=16, 16>15) 15*2^4-1=239 which is prime. [URL="http://www.rieselprime.de"]www.rieselprime.de[/URL] lists primes when 2^n<k, even though these aren't technically Riesel numbers.
[quote=Primeinator;181562]Finally, n = k is also a possibility. 3*2^3 -1 = 23 = ! prime.[/quote]This might interest you: n*2^n+1 is a [URL="http://mathworld.wolfram.com/CullenNumber.html"]Cullen Number[/URL]. (essentially a Proth number with k=n and no 2^n>k restriction; the Proth side equivalent of a Woodall number)
[quote=Primeinator;181562]Apparently, all three scenarios are possible?[/quote]Apparently.

So...
When k*2^n+1 is prime, k*2^n+1 is called a Proth prime
When k*2^n-1 is prime, k*2^n-1 is called a Riesel prime
When k*2^n+1 is composite for every n with this specific k, k is called a Sierpinski number
When k*2^n-1 is composite for every n with this specific k, k is called a Riesel number
When k*2^n+1 with odd k, positive integer n, and 2^n>k, k*2^n+1 is called a Proth number
When k*2^n-1 with odd k, positive integer n, and 2^n>k, k*2^n-1 is called ...what? (we're referring to it as Riesel number here, but that's technically incorrect since that refers to the equiv. of a Sierpinski number)

or in text: "Riesel number" technically refers to a k such that all k*2^n-1 are composite, and "Riesel prime" refers to primes of the form k*2^n-1, right? Is there any name for numbers of the form k*2^n-1, analogous to "Proth number" for numbers of the form k*2^n+1? I know there is rarely confusion, at least in projects that aren't searching for Riesel numbers, but it is still an incorrect and vague reference.

R.D. Silverman 2009-07-19 19:14

[QUOTE=Primeinator;181504]How would 2^n < k in of itself make a number riesel? I thought that any number of the form k*2^n -1 is a riesel number.[/QUOTE]

[b]Every[/b] single integer is of the form k*2^n - 1.

Mini-Geek 2009-07-19 23:14

[quote=R.D. Silverman;181758][B]Every[/B] single integer is of the form k*2^n - 1.[/quote]
Is this assuming k and n are integers? If so, I don't see how this could be.


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