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## Proving Sierpinski conjecture before SoB completes its task

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• Each Sierpinski (and Riesel) number and the dual of it always have the same covering set as it is easy to see. So we only need to find a prime for a candidate
Message 1 of 7 , May 17, 2008
Each Sierpinski (and Riesel) number and the dual of it always have the
same covering set as it is easy to see.

So we only need to find a prime for a candidate in any one of the
forms to eliminate it from the other side.

Looking at the remaining candidates in SoB and in the dual Sierpinski
search of Payam Samidoost we see that there is no number belonging to
both lists, so no one of the 6 remaining candidates at SoB can be a
solution to the problem, and the same to Payam's as well.

In fact, the only two common candidates when Payam launched his
project were 19249 and 28433.

Payam found a prime for the dual of 19249 on August 17, 2002 so when
an anonymous member of TeamPrimeRib submitted a prime for 28433 on
December 30, 2004 to SoB, the search could have been called quits.

Lélio
• From: Lélio Ribeiro de Paula ... By Sierpinski s definitions, finding a prime in the dual set does not remove the k value as a candidate to be what was later
Message 2 of 7 , May 18, 2008
From: Lélio Ribeiro de Paula
> Each Sierpinski (and Riesel) number and the dual of it
> always have the
> same covering set as it is easy to see.
>
> So we only need to find a prime for a candidate in any one
> of the
> forms to eliminate it from the other side.
>
> Looking at the remaining candidates in SoB and in the dual
> Sierpinski
> search of Payam Samidoost we see that there is no number
> belonging to
> both lists, so no one of the 6 remaining candidates at SoB
> can be a
> solution to the problem, and the same to Payam's as
> well.
>
> In fact, the only two common candidates when Payam launched
> his
> project were 19249 and 28433.
>
> Payam found a prime for the dual of 19249 on August 17,
> 2002 so when
> an anonymous member of TeamPrimeRib submitted a prime for
> 28433 on
> December 30, 2004 to SoB, the search could have been called
> quits.

By Sierpinski's definitions, finding a prime in the dual set does not remove the k value as a candidate to be what was later termed a Sierpinski number.

Phil
• ... Indeed, the prime of the form k+2^n might be in the covering set for the numbers of the form k*2^n+1.
Message 3 of 7 , May 18, 2008
Phil Carmody wrote:
>
> By Sierpinski's definitions, finding a prime in the dual set does not remove the k value as a candidate to be what was later termed a Sierpinski number.
>

Indeed, the prime of the form k+2^n might be in the covering set for the numbers
of the form k*2^n+1.
• Your idea is very interesting. My readings about near-repdigit numbers has made me fascinated by Sierpinski and Riesel numbers and especially their covering
Message 4 of 7 , May 18, 2008
Your idea is very interesting.

My readings about near-repdigit numbers has made me fascinated by
Sierpinski and Riesel numbers and especially their covering sets.

If you look at the search of Payam Samidoost, can we prove any of the
sixty-three or so possible candidates below the smallest provable
Riesel number 509203 are not Riesel numbers. I would be especially
interested in 2293 and 9221 because they are so small.

--- In primenumbers@yahoogroups.com, Lélio Ribeiro de Paula
<lelio73@...> wrote:
>
> Each Sierpinski (and Riesel) number and the dual of it always have the
> same covering set as it is easy to see.
>
> So we only need to find a prime for a candidate in any one of the
> forms to eliminate it from the other side.
>
> Looking at the remaining candidates in SoB and in the dual Sierpinski
> search of Payam Samidoost we see that there is no number belonging to
> both lists, so no one of the 6 remaining candidates at SoB can be a
> solution to the problem, and the same to Payam's as well.
>
> In fact, the only two common candidates when Payam launched his
> project were 19249 and 28433.
>
> Payam found a prime for the dual of 19249 on August 17, 2002 so when
> an anonymous member of TeamPrimeRib submitted a prime for 28433 on
> December 30, 2004 to SoB, the search could have been called quits.
>
> Lélio
>
• ... No, it cannot. The covering sets of all known Riesel and Sierpinski numbers are exactly the same as that of their duals, as can be easily seen. So if a
Message 5 of 7 , May 19, 2008
Jack Brennen wrote:
>
> Indeed, the prime of the form k+2^n might be in the covering set for
> the numbers of the form k*2^n+1.
>

No, it cannot.

The covering sets of all known Riesel and Sierpinski numbers are
exactly the same as that of their duals, as can be easily seen.

So if a prime for a dual of a Sierpinski number were to be in the
covering set of that particular Sierpinski number, it should also be
in the dual covering set as well, which contradicts the definition of
covering sets.

Lélio
• ... Now reread what Jack wrote (which was also going be in my original post too, but thinking that it was a bit obvious I removed it for brevity), and think a
Message 6 of 7 , May 19, 2008
--- On Mon, 5/19/08, Lélio Ribeiro de Paula <lelio73@...> wrote:
> Jack Brennen wrote:
> > Indeed, the prime of the form k+2^n might be in the
> covering set for
> > the numbers of the form k*2^n+1.
>
> No, it cannot.
>
> The covering sets of all known Riesel and Sierpinski
> numbers are
> exactly the same as that of their duals, as can be easily
> seen.
>
> So if a prime for a dual of a Sierpinski number were to be
> in the
> covering set of that particular Sierpinski number, it
> should also be
> in the dual covering set as well, which contradicts the
> definition of
> covering sets.

Now reread what Jack wrote (which was also going be in my original post too, but thinking that it was a bit obvious I removed it for brevity), and think a bit more.

Phil
• ... I tried (in private email) giving the example of the dual sequences: 10^n-7 7*10^n-1 They both have the same covering set (it can be found in under a
Message 7 of 7 , May 19, 2008
Phil Carmody wrote:
>
> Now reread what Jack wrote (which was also going be in my original post too, but thinking that it was a bit obvious I removed it for brevity), and think a bit more.
>

I tried (in private email) giving the example of the dual sequences:

10^n-7
7*10^n-1

They both have the same covering set (it can be found in under a
minute with just a little thought). One of the sequences has a
very easy to find prime; the other one is easily proven to have no
primes.
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