[seqfan] Re: C.Boyd Re: More composite numbers are needed for A226181 (by way of moderator)

[Max Alekseyev]

Charles, this is a great idea. I've tested all 2-pseudoprimes below
2^64 and found only three new terms:

54745942917121 = 13365708720*2^12 + 1
51125767490519041 = 12481876828740*2^12 + 1
18314818035992494081 = 69865486282320*2^18 + 1

I've also found one new large term:
32768*2^16369 + 1

Regards,
Max

On Tue, May 27, 2014 at 2:33 PM, Charles Greathouse
<charles.greathouse at case.edu> wrote:
> If they are 2-pseudoprimes (as Joni Teräväinen says) then you should be
> able to use Feitsma's enumeration to 2^64 to extend the search considerably.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
>
> On Tue, May 27, 2014 at 11:10 AM, Olivier Gerard
> <olivier.gerard at gmail.com>wrote:
>
>> From: "C Boyd" <cb1 at gmx.co.uk>
>> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
>> Date: Tue, 27 May 2014 15:02:29 +0200
>> Subject: Re: [seqfan] More composite numbers are needed for A226181
>>
>> Dear SeqFans,
>>
>> I have used Pari to identify all relevant composites below 10^11.
>> As far as I can tell, all but one (43796171521) have been
>> previously discovered either by the A226181 originator Lear Young,
>> by Max Alekseyev in this thread, or by Peter Kosinar in
>> <
>>
>> http://mathoverflow.net/questions/168045/are-all-counterexamples-of-oeis-a226181-both-poulet-numbers-and-proth-numbers
>> >.
>>
>> In tabular form, with annotation for Proth number status, the
>> complying composites < 10^11 are:
>>
>> Y/N = Proth/not Proth
>> Y 12801       = 2^9 * 25 + 1         = 512 * 25 + 1          Lear Young
>> Y 348161      = 2^12 * 85 + 1        = 4096 * 85 + 1         Lear Young
>> Y 3225601     = 2^11 * 1575 + 1      = 2048 * 1575 + 1       Lear Young
>> Y 104988673   = 2^17 * 801 + 1       = 131072 * 801 + 1      Max Alekseyev
>> Y 4294967297  = 2^32 + 1             = 4294967296 + 1        Max Alekseyev
>> N 7816642561  = 2^15 * 238545 + 1    = 32768 * 238545 + 1    Peter Kosinar
>> N 43796171521 = 2^8 * 171078795 + 1  = 256 * 171078795 + 1   CB
>> N 49413980161 = 2^15 * 1507995 + 1   = 32768 * 1507995 + 1   Peter Kosinar
>>
>> Max's "power of 2" numbers are the Fermat numbers > 65537. If
>> there are an infinite number of composite Fermat numbers (as seems
>> likely), each of them is a Proth number fulfilling the original
>> conditions.
>>
>> CB
>>
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