• ## Relatively major problem ripe for expansion

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• A115091 in the OEIS declares there are no primes 613
Message 1 of 4 , Feb 7, 2011
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A115091 in the OEIS declares there are no primes 613<p<10^6 such that there is a k<p for which p^2|k!+1 and no p<10^8 with k<1201. Three PARI/GP windows are going to easily push the first limit to 10^7 (or find a solution) in about 7 weeks. If anybody with real programming capacity is interested, it wouldn't require very much to put the whole problem at the p>10^8 level.
On another note, I haven't heard back from Jens on an offer of sharing credit for a nice titanic curio I put in his hands for proof, leading me to believe he's working on it. I don't suppose there's likely to be any harm done by stating it now, since he would have had quite a number of hours with it. The first titanic near-repunit prime with a near-repunit prime number of digits, (10^1117-1)/9+4*10^92, is the 101st such prime (of any size), assuming no rare ispseudoprime false positive and defining near-repunits to necessarily have at least 3 digits.
Jim Merickel
• ... You mailed me less than a day ago. Following up on your work is not all I do, and I don t want to share a curio just because I verified it afterwards.
Message 2 of 4 , Feb 7, 2011
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James Merickel wrote:
> On another note, I haven't heard back from Jens on an offer of sharing
> credit for a nice titanic curio I put in his hands for proof, leading me to
> believe he's working on it. I don't suppose there's likely to be any harm
> done by stating it now, since he would have had quite a number of hours with
> it. The first titanic near-repunit prime with a near-repunit prime number
> of digits, (10^1117-1)/9+4*10^92, is the 101st such prime (of any size),
> assuming no rare ispseudoprime false positive and defining near-repunits to
> necessarily have at least 3 digits.

You mailed me less than a day ago. Following up on your work is not all I do,
and I don't want to share a curio just because I verified it afterwards.
However, after seeing your post here I have confirmed it is the 101st prp.
Marcel Martin's Primo has proved the first 77 prp's and the last.
I skipped the remaining 23 which have 811 and 911 digits:
(10^811-1)/9+6*10^14
(10^811-1)/9+6*10^82
(10^811-1)/9+4*10^115
(10^811-1)/9+6*10^155
(10^811-1)/9+7*10^216
(10^811-1)/9+7*10^242
(10^811-1)/9+7*10^388
(10^811-1)/9+7*10^406
(10^811-1)/9+4*10^480
(10^811-1)/9+6*10^530
(10^811-1)/9+4*10^577
(10^811-1)/9+1*10^600
(10^911-1)/9+8*10^327
(10^911-1)/9+6*10^357
(10^911-1)/9+6*10^402
(10^911-1)/9+5*10^465
(10^911-1)/9+2*10^475
(10^911-1)/9+6*10^707
(10^911-1)/9+5*10^717
(10^911-1)/9+8*10^732
(10^911-1)/9+6*10^792
(10^911-1)/9+6*10^795
(10^911-1)/9+2*10^890

It might take around half a GHz day to verify them with Primo from
http://www.ellipsa.eu/public/primo/primo.html

The 23 input files can quickly be made with
http://users.cybercity.dk/~dsl522332/math/certif/primoin.zip

--
Jens Kruse Andersen
• As I explained (and apolgized for) to Mr. Andersen, although I correctly assumed he read my message quite a while before, I was incorrect in assuming the only
Message 3 of 4 , Feb 8, 2011
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As I explained (and apolgized for) to Mr. Andersen, although I correctly assumed he read my message quite a while before, I was incorrect in assuming the only likely possibilities were a lack of interest--generating a brief reply to that effect--or that he had already started some work on it, assumptions based on a sense they formed a dichotomy with two quick actions ignoring the possibility he filed it in memory and went about other activities. Anyway, silly of me.
The 23 remaining cases all went through PARI/GP isprime tests successfully, and it's as much a mystery to me as I'm sure it would be to anyone that I could run twice as many (at least) programs as put my cpu at 100% usage supposedly, but it's done.

On Mon Feb 7th, 2011 7:41 PM EST Jens Kruse Andersen wrote:

>James Merickel wrote:
>> On another note, I haven't heard back from Jens on an offer of sharing
>> credit for a nice titanic curio I put in his hands for proof, leading me to
>> believe he's working on it. I don't suppose there's likely to be any harm
>> done by stating it now, since he would have had quite a number of hours with
>> it. The first titanic near-repunit prime with a near-repunit prime number
>> of digits, (10^1117-1)/9+4*10^92, is the 101st such prime (of any size),
>> assuming no rare ispseudoprime false positive and defining near-repunits to
>> necessarily have at least 3 digits.
>
>You mailed me less than a day ago. Following up on your work is not all I do,
>and I don't want to share a curio just because I verified it afterwards.
>However, after seeing your post here I have confirmed it is the 101st prp.
>Marcel Martin's Primo has proved the first 77 prp's and the last.
>I skipped the remaining 23 which have 811 and 911 digits:
>(10^811-1)/9+6*10^14
>(10^811-1)/9+6*10^82
>(10^811-1)/9+4*10^115
>(10^811-1)/9+6*10^155
>(10^811-1)/9+7*10^216
>(10^811-1)/9+7*10^242
>(10^811-1)/9+7*10^388
>(10^811-1)/9+7*10^406
>(10^811-1)/9+4*10^480
>(10^811-1)/9+6*10^530
>(10^811-1)/9+4*10^577
>(10^811-1)/9+1*10^600
>(10^911-1)/9+8*10^327
>(10^911-1)/9+6*10^357
>(10^911-1)/9+6*10^402
>(10^911-1)/9+5*10^465
>(10^911-1)/9+2*10^475
>(10^911-1)/9+6*10^707
>(10^911-1)/9+5*10^717
>(10^911-1)/9+8*10^732
>(10^911-1)/9+6*10^792
>(10^911-1)/9+6*10^795
>(10^911-1)/9+2*10^890
>
>It might take around half a GHz day to verify them with Primo from
>http://www.ellipsa.eu/public/primo/primo.html
>
>The 23 input files can quickly be made with
>http://users.cybercity.dk/~dsl522332/math/certif/primoin.zip
>
>--
>Jens Kruse Andersen
>
• ... I d be surprised if it wasn t either known or trivially checkable way beyond that limit. I know I ve pulled out lots of factors out of k!+1, all that needs
Message 4 of 4 , Feb 8, 2011
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--- On Mon, 2/7/11, James Merickel <merk7777777@...> wrote:
> A115091 in the OEIS declares there
> are no primes 613<p<10^6 such that there is a k<p
> for which p^2|k!+1 and no p<10^8 with k<1201.

I'd be surprised if it wasn't either known or trivially checkable way beyond that limit. I know I've pulled out lots of factors out of k!+1, all that needs to be done is to check that they're not double factors. I don't think I was the central resource for those factors, maybe it was redgolpe who was, I forget now.

> Three PARI/GP windows are going to easily push the first
> limit to 10^7 (or find a solution) in about 7 weeks.
> If anybody with real programming capacity is interested, it
> wouldn't require very much to put the whole problem at the
> p>10^8 level.

Just from the data I've got here, there are no squared factors of k!+/-1 for k<10000 and p<1483562771179

\$ tail -n 1 factorialsmall.fac
(3233!+1)%1483562771179

\$ sed -e s/\$/^2/ < factorialsmall.fac | gp -q | grep '= 0'
[silence]

I have other logs of other runs, but there are gaps. That's why I don't think I was the one keeping a definitive record.

Phil

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