- --- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:
>

Yes, Jack, you are of course quite right. Thanks for pointing that out so neatly.

> Isn't this (for example) a much easier way to show that 10223

> has no such covering set:

>

> 10223*2^137+1 ==

> 1904660910466121 * 935125926286211318869570932217

>

> Implying of course that any covering set for 10223 includes

> one of those two primes...

>

Stepping back from the code, I realized quite soon after posting that all it did was find, for each k, any n value that gave a large smallest factor, and that is achievable with lots of off-the-shelf sieving and factorizing programs.

Mike - Jack Brennen wrote:
> 10223*2^137+1 ==

A Google search of 1904660910466121 finds

> 1904660910466121 * 935125926286211318869570932217

http://www.math.sc.edu/~filaseta/papers/SierpinskiEtCoPapNew.pdf

They reached the same factorization as Jack and listed:

n, smallest prime factor of 10223*2^n+1

77 619033

101 45677096693

137 1904660910466121

The next n with no factor below 10^9 is 509.

What is the smallest factor of 10223*2^509+1?

It is probably above 10^30.

--

Jens Kruse Andersen - --- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:
>

In defence of my code, it solved all 6 cases in 15 mins, without my having to depend on referring to web pages of known factorisations, which are the result of significant computational effort, and which you perhaps used to find that n=137 index among the the prima facie hundreds to choose from?

> Isn't this (for example) a much easier way to show that 10223

> has no such covering set:

>

> 10223*2^137+1 ==

> 1904660910466121 * 935125926286211318869570932217

>

> Implying of course that any covering set for 10223 includes

> one of those two primes...

>

Mike - No significant computational effort needed for this case. PARI code:

k=10223;for(i=1,200,f=factorint(k*2^i+1,15);if(length(f[,1])==1,print([k,i,factorint(k*2^i+1)])))

Runs with elapsed time about 2 seconds. Output:

[10223, 77, [619033, 1; 2495595681878097381929, 1]]

[10223, 101, [45677096693, 1; 325387086953, 1; 1743849966293, 1]]

[10223, 137, [1904660910466121, 1; 935125926286211318869570932217, 1]]

It's also quick to find one for 22699 which has no small factors:

[22699, 190, [84884846681, 1;

419638892754915061028443044698201980071935256023017, 1]]

Unfortunately, the other 4 candidates don't present such

low-hanging fruit; their k*2^n+1 numbers either have small

factors, or they are too big to factor quickly.

mikeoakes2 wrote:>

>

> --- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:

>> Isn't this (for example) a much easier way to show that 10223

>> has no such covering set:

>>

>> 10223*2^137+1 ==

>> 1904660910466121 * 935125926286211318869570932217

>>

>> Implying of course that any covering set for 10223 includes

>> one of those two primes...

>>

>

> In defence of my code, it solved all 6 cases in 15 mins, without my having to depend on referring to web pages of known factorisations, which are the result of significant computational effort, and which you perhaps used to find that n=137 index among the the prima facie hundreds to choose from?

>

> Mike

>

>

>

> ------------------------------------

>

> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com

> The Prime Pages : http://www.primepages.org/

>

> Yahoo! Groups Links

>

>

>

>

>