-- Forwarded message --
Date: Mon, 20 Sep 1999 19:55:53 +0200 (DFT)
From: Wilfrid Keller <[EMAIL PROTECTED]>
To: [EMAIL PROTECTED]
September 20, 1999
Dear Warut, Dear Colleagues:
Concerning the subject of Mersenne numbers M(q) = 2^q - 1, where
q = 2^p - 1 itself is a Mersenne prime, I am sorry I hadn't seen
your relevant web page before. As the factor of M(M(13)) that
I had found in 1976 might suggest, I have shown interest in this
particular kind of Mersenne numbers since many, many years. Re-
garding the factor of M(M(31)) just rediscovered by Lucas Wiman,
I regret to inform you that it was already "published" in 1983.
Unfortunately, Guy Haworth's notes (please see the references be-
low) were released only "privately", but were in fact widely cir-
culated. The second prime factor of M(M(31)), found by Tony
Forbes, was also known to me for some years (again, see below).
Let me take this opportunity to communicate to you the complete
records of my search for factors of numbers M(M(p)), particular-
ly including the attained limits, which may help to avoid further
duplication. Also, please note that M(M(127)) was shown to have
no factor 2h x M(127) + 1 for h < 6.8 x 10^8.
If you should find it of interest to forward my data to some per-
tinent mailing list (NMBRTHRY or so), please feel free to do so.
And if you have any questions related to this topic, I would be
glad to respond.
With my best wishes to all of you,
Wilfrid Keller
[PS: This note is by no means intended to earn the money allegedly
offered for a proof that M(M(31)) is not a prime. Anyway,
the reward would have been for discovering a factor, and not
for giving a reference where to look it up, I suppose.]
Known prime factors 2h*M(p) + 1 of
"iterated" Mersenne numbers M(M(p))
as of November 1996
ph DiscovererReferences
1320644229Keller Haworth [1983],
Ribenboim [1988]
17 884RobinsonRobinson [1957]
17 245273Keller Haworth [1983]
19 60RobinsonRobinson [1957]
19 5480769Keller Found Aug 20, 1994
(unpublished)
31 68745Keller Haworth [1983]
3120269004Keller Found Aug 28, 1994
(unpublished)
References
--
Raphael M. Robinson, Some factorizations of numbers of the
form 2^n +/- 1, MTAC (Math. Comp.) 11 (1957), 265-268.
Guy Haworth, Mersenne Numbers, Reading, Berkshire, 1983
(privately published notes), and subsequent updates.
quoted as reference #203 in
Daniel Shanks, Solved and Unsolved Problems in Number
Theory, 3rd ed., Chelsea, New York 1985,
and as reference #223 in
John Brillhart, D.H. Lehmer, J.L. Selfridge, Bryant
Tuckerman, and S.S. Wagstaff, Jr., Factorizations of
b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, 2nd
ed., American Mathematical Society, Providence, Rhode
Island, 1988.
Paulo Ribenboim, The Book of Prime Number Records, Springer,
New York 1988, p. 80.
Search limits h < L(p) for factors of
"iterated" Mersenne numbers M(M(p))
as of November 1996
pL(p)
13 6.7 x 10^8
17 3.3 x 10^8
19 4.0 x 10^8
31 3.1 x 10^8
61 7.5 x 10^8
89 5.3 x 10^8
107 5.2 x 10^8
127 6.8 x 10^8
521 3.6 x 10^5
607 3.4 x 10^5
_
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