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

Olivier Gerard olivier.gerard at gmail.com
Tue May 27 17:10:05 CEST 2014


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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