Problems & Puzzles:
Problems
Problem 29. Brier Numbers
A Sierpinski
Number is an integer k such that k.2^{n}+1 is composite
for any integer value of n. In 1962, John Selfridge discovered the Sierpinski
number k = 78557, which is now believed to be in fact the smallest
such number.
A Riesel
Number is an integer k such that k.2^{n}1 is composite
for any integer value of n. In 1956 Riesel showed that k = 509203 has this property.
Two months
ago I asked myself if it could exists an integer k such that at the
same time k.2^{n}+1 and k.2^{n}1 are composite for
any value of n.
Quickly
faced with the crude fact that I had not any idea about how to answer
my own question J …
then in a very stylish manner I simply switched to ask to my friend Wilfrid Keller the same question.
His answer
arrived almost immediately: that kind of numbers not only should exist
but very recently (?) Eric
Brier produced the first one known. Let me call this kind of
numbers “Brier Numbers”.
Since
28/9/98 and up today, 16/19/99, the smallest
know Brier Number is 29364695660123543278115025405114452910889,
that I will call it “The
smallest known Brier number” or simply
SKB(28/9/98)
SKB(28/9/98,
E. Brier) = 29364695660123543278115025405114452910889
(41
digits)
Questions:
 Can
you produce a smaller Brier Number than SKB(28/9/98)?
 Can
you estimate the magnitude order of the smallest Brier Number
(that I will call The AlphaBrier Number or simply B_{a}?
Hints:
After the Keller’s kind
communication I got in touch with Eric
Brier asking him for the method employed by him to produce this
kind of numbers. I
offered him also my pages for publishing his method and for exposing
all the other open questions that should arise around the Brier
Numbers.
Please click here
to download the document that Eric
produced and that must help you to organize your own search. This
document also poses several other open questions that “arise quite
naturally” and two complements.
I have added other two
complements:
a)
A letter
from Wilfrid Keller to Chris
Nash, Eric Brier & Yves Gallot, dated the 26/9/98 that in the
middle of many other things suggests how to get a smaller Brier number
than the existing at that time.
b)
The announcement from Eric
Brier about his discovery of SKB(28/9/98)
Yves
Gallot has discovered (15/01/2000) a smaller Brier number. This
one is 30 digits large against the previous record 41 digits large
obtained by Eric Brier the 28/9/98. This is the Gallot's
announcement:
"Dear Eric, Carlos and Wilfrid,
I just found a smaller Brier number:
623506356601958507977841221247 (30 digits)
The sets are: { 3, 7, 73, 13, 19, 241, 37, 109, 97, 673 } with e1
= 144 = 2^4*3^2 and { 3, 5, 17, 257, 65537, 641, 6700417 } with e2 = 64
= 2^6 then for the complete set e = 576 = 2^6 * 3^2
I evaluated k with Chinese Remainder Theorem for 64*576
permutations to find the smallest solution. I wrote a program to search
for Sierpinski/Riesel sets with a fixed value for e. I excluded the
primes 5, 17, 257, 65537, 641, 6700417 of the sets and searched the
smallest solution. I will continue the search by trying some other
"exclusions".
I would like to thank you for your discoveries, complete Web pages
and theoretical presentation of Sierpinski/Riesel/Brier numbers. I would
have not be able to write my program without them! Best wishes, Yves"
So, now the smallest known Brier number to beat is:
SKB(15/1/2000, Y. Gallot) = 623506356601958507977841221247
(30 digits)
Minutes after receiving his email I asked to Gallot the
following: "One question:how are you deciding "the
exclusions"? following certain rules? a combination of rules &
random?". His answer arrived one hour later:
"I searched a set which generates a Riesel number by
excluding 5, 17, 257,
65537, 641, 6700417 because it is known that { 3, 5, 17, 257, 65537,
641, 6700417 } generates a Sierpinski number (and 3 can be used in both
sets). The idea is to:
1  find a "small" set of prime that generates a Riesel
number.
2  search a set of prime that doesn't include the set of primes
previously found, except 3, and generates a Sierpinski number.
Of course, the union of the two sets generates a Brier number. The
idea is just Eric's method, but I automatized this process and the first
run of the program (still under construction and today semiautomatic)
was successful. I used no special rules but a bruteforce algorithm that
generates all cases for e=24 to .... Eric found a solution with e1=96
and e2=288. My program found that e1=64 and e2=144 are enough. It also
rediscovered Eric's solution.
I hope to finish my program next week and maybe a smallest
solution will be found. If it can check all solutions for e1=24 to 516
and for e2=24 to 516, we have a good chance to find the smallest
"cyclic" Brier number in it..."
***
Comment: The before record remained one year and 3 months unbeaten. I
wonder how long this new record will remain...and if the conjecture of
Chris Nash (See Problem 30) about a possible Brier number of 1011 digits
will be confirmed by the method used by Brier and Gallot...
***
Ink was still fresh when the following message from Gallot arrived:
"Wilfrid, Carlos and Eric, I just found a new smaller Brier
number:
3872639446526560168555701047 (28 digits)
The sets are:{ 3, 7, 73, 13, 19, 241, 37, 109, 97, 673 } with e1 = 144
= 2^4 * 3^2 and { 3, 5, 31, 17, 11, 151, 41,
331, 61681, 61 } with e2 = 120 = 2^3 * 3 * 5 then for the complete set e =
720 = 2^4 * 3^2 * 5. I evaluated k with Chinese Remainder Theorem for
2*2880*1152 permutations to find the smallest solution. I continue the
search...Yves"
SKB(16/1/2000, Y. Gallot) = 3872639446526560168555701047
(28 digits)
I guess that it's a good idea to stop any kind of
forecasting about remaining time of records for the moment...:)
***
Yeaph!...it was a good idea. The morning after I had the third
and smaller Brier number by Yves Gallot in my inbox. This is his
email:
"Last night, my computer found a smaller Brier
number:878503122374924101526292469 (27 digits)
The sets are:{ 3, 7, 73, 13, 257, 19, 241, 37, 109, 97 } with e1 = 144 =
2^4 * 3^2 and{ 3, 5, 31, 17, 11, 151, 41, 331, 61681, 61 } with e2 = 120 = 2^3
* 3 * 5 then for the complete set e = 720 = 2^4 * 3^2 * 5.
k was evaluated with Chinese Remainder Theorem for 2*2880*1152
combinations. The product of the primes of the
two sets is:9174644185821387 * 670447226147431755 / 3 =
2050371581757870446479551733981395. This 34 digit number is probably the
smallest possible one. But I still have some candidates that are a bit larger
and may generate a smaller k, but we should be closed to the minimal for k."
***
So the new smaller known Brier numbers is:
SKB(17/1/2000, Y. Gallot) = 878503122374924101526292469
(27 digits)
***
This Monday (17) I received two more letters highly stimulated
by the Gallot's results. The first one came from Wilfrid Keller and the
second one from Chris Nash. I believe the both letters should be
known completely, specially if you are interested in producing smaller and
smaller Brier numbers. Both of the letters provide interesting clues to
organize the future search by the method currently employed by Brier
& Gallot. In particular the Keller's letter provides some new
results to the search approach suggested in the Problem 30 for finding the Brieralpha
number ...
I have merged the two letters in one txt document. Click here
to download it.
***
Yves Gallot has wrote in his own site a note
about his search for Brier numbers that maybe will be of interest to Brier
numbers hunters.
***
Ten Years later (May 2010), Emmanuel Vantieghem wrote:
The number k = 47867742232066880047611079 is a Brier
number of 26 digits. According to the text of problem 29, I guess it is
the smallest known at present.
A covering set for the sequence ( k 2^n1 ) is S1 =
{3,7,11,19,31,37,41,61,73,109,151,331} and a covering set for the
sequence ( k 2^n+1 ) is S2 = {3,5,13,17,97,241,257}. I was brought to
this in attempting to prove the assertion in problem 52. Indeed, for
every n, at least one of the primes of S1 divides k2^n and at
least one of the primes of S2 divides k+2^n, as can easily be
checked. Considering not only positive values for n but also negative
ones lead me to my result.
The evidence of my claim lies in the covering sets
S1 and S2. Anyone can check that for any value of n, the
number k 2^n  1 is divisible by at least one element of the set S1.
It suffices to verify this for n = 1, 2, ..., 180, since the appearance
of a prime in the sequence ( k 2^n  1 ) is periodic. The periods of
the set S1 are indeed {2,3,10,18,5,36,20,60,9,36,15,30} with LCM
180.
The same holds for k 2^n + 1 and S2. The periods for the primes of
S2 are {2,4,12,8,48,24,16} with LCM 48.
So,
SKB(26/5/2010, E. Vantieghem) =
47867742232066880047611079
(26 digits)
***
On 28/05/2010, I sent an email to Mr. Wilfrid Keller, Yves Gallot & Eric Brier, in order to let them know the Emmanuel's result.
Immediately Yves responded:
The
24digit Brier number 143665583045350793098657 was
reported in 2007:
"On
Powers Associated with Sierpinski Numbers, Riesel Numbers and Polignac’s
Conjecture", Michael
Filaseta, Carrie Finch, Mark Kozek.
http://www.math.sc.edu/~filaseta/papers/SierpinskiEtCoPapNew.pdf
It was
a real surprise to me because I really thought that I had found the
smallest Brier during my search in 2000. My strategy of “best” and “good
“sets was not optimal. Now, I’m working on different projects and never
took the time to extend the
systematic generation of covering sets and to check if the 24digit
Brier number can be obtained by combining them with the power of today’s
computer… may be a sort of “computational proof” that the smallest known
Brier number is optimal is something possible today.
***
So, the ball is rolling on... and,
SKB(2007, Filaseta et al) =
143665583045350793098657
(24 digits)
***
See also:
http://mathworld.wolfram.com/BrierNumber.html
Sloane, N. J. A. Sequence A076335 in
"The OnLine Encyclopedia of Integer Sequences."
In particular the following line:
"There
are no Brier numbers below 10^9. [From Arkadiusz Wesolowski (math(AT)wesolowski.ids.pl),
Aug 03 2009]"
This means that the smallest Brier numbers is somewhere in
bewteen 10^9 and
143665583045350793098657.
Does somebody has the A. Wesolowski
proof of his assertion?
***
Status Table 
date 
Author 
Brier Number 
Digits 
28sep98 
Eric Brier 
29364695660123543278115025405114452910889 
41 
15ene00 
Yves Gallot 
623506356601958507977841221247 
30 
16ene00 
Yves Gallot 
3872639446526560168555701047 
28 
17ene00 
Yves Gallot 
878503122374924101526292469 
27 
26may10 
E. Vantieghem 
47867742232066880047611079 
26 
2007 
Filaseta et al 
143665583045350793098657 
24 
Next? 
Next? 
Next? 
Next? 
***
Regarding my question "Does somebody has the A. Wesolowski
proof of his assertion?" posed above last week I received two answer,
in this order:
a) W. Keller wrote ():
...I had also been aware of the paper by Filaseta, Finch
and Kozek. But I didn't think about relating
it to your "Problem 29". Your readers might be interested in knowing
where this paper
has finally been published:
Journal of Number Theory, vol. 128 (2008), pp. 19161940;
electronic: April 2008.
I also wanted to make some remarks concerning your note about
Arkadiusz Wesolowski's verification that "There are no Brier
numbers below 10^9".
We seem to know considerably more than that. Looking at the
table on your page on "Problem 49. Sierpinskilike numbers",
we see that k=2152690373 is given as the "smallest uncertain
candidate" for being a Brier number (but actually
2152690373*2^22461+1 is a prime, so this k really isn't such
a "candidate" anymore).
Revising our earlier correspondence, I found the following
additional information. On February 14, 2000 I copied to you:
I have a new lower bound for the smallest Brier
number: it must
be > 10^9 !
[...]
Here is the complete list of "smallest" primes having n >= 2^14,
for k < 10^9 :
16935761 22394
100604513 41422
102017081 17419+
118373279 82587+
188001043 30554+
257305073 18825+
270704167 85461
282681079 58783
308009629 16742+
343689013 58314+
411390743 18284
418452299 22648
585540229 24437
615516221 37854
623603909 19989+
630511667 16587+
637741513 78920+
753118759 18529
807512089 27626+
820235483 17004
835116281 20311+
840923429 16473+
870834563 38990
920892347 22452
927795317 17135+
This list might be seen as an indication of a "proof of his
assertion."
In March 2001 I told you that I was about to completing the
search for all k < 2^31 = 2147483648, finally reaching the
abovementioned limit of k = 2152690373.
In the meantime you had been organizing a coordinated search
for k > 2^31 at
http://www.primepuzzles.net/private/index.htm .
On April 24, 2002 I wrote to you, summarizing the results of
our joint efforts at that time:
Esta vez tengo noticias acerca del coeficente
k=1355477231
relacionado con el problema numero 30: acabo de enviar el
numero primo 1355477231*2^356981 + 1 (107472 digitos) a la
lista de Chris Caldwell! Ademas he probado que los corres
pondientes numeros 1355477231*2^n  1 son compuestos para
todo n < 393000.
A traves de este resultado ahora sabemos que no existe nin
gun "numero de Brier" menor que k=2294020991, el "candi
dato" identificado por usted.
About this coefficient k your "Status of the search for
k=2294020991 the 24/04/2004" said that the range n=790000800000
was still "Working". But now, to my great surprise, I see on Chris
Caldwell's database the prime 2294020991*2^800493+1, submitted on
December 17, 2004!
According to your "Ranges proposed & status of search", wouldn't
this imply that there is no Brier number below k=2362690377, at
least?
Next: What about the two "survivors" (only two?) in the interval
2362690377 <= k <= 2572690377 with no primes for n <= 90000 ?
Which are these, and have they been searched further?
Aren't we possibly "done" up to k = 3622690387, altogether?
I think it would be really interesting to continue this search
until a really "hard" candidate might be identified.
In any case, on the page on "Problem 30" you could probably add
the prime 2294020991*2^800493+1 as the next step of the
"stepladder".
Finally, I wanted to give you a (quite belated) update to your
"Problem 31": In 2004, Maxim Vsemirnov has slightly improved
John Nicol's record (near the end of your page), please see the
attached paper
b) A. Wesolowski wrote:
Through a computer search I noted that the number is
greater than 10^9.
My strategy was simple:
find k such that k.2^n + 1 and k.2^n  1 are both
composite for all n up to 1000.
* The point is that the trial division method is
helpful only if n > 1000.
5 days from k = 10^7 to k = 10^9 is amazing!
I generated a list of k's. Next, I did change one
thing (*).
After the tests were done, there was a proof.
The computing continues as the rules I presented
above,
k = 2335574321, exponent = 10799+
***
But the most surprising thing was that A. Wesolowski
reported that the SKB(26/5/2010, E. Vantieghem) =
47867742232066880047611079
(26 digits) was already known and calculated as part of another search in
1975!!!:
This number was found out in 1975.
From this article I take out this image:
***
BTW, this number was also the core of our
Problem 52
and the declared origin of the search of Mr. Emmanuel Vantieghem, as he
stated in his email (I
was brought to this in attempting to prove the assertion in problem 52).
So, accordingly and is my interpretation, that Mr. Vantieghem took this
number from the Problem 52 and noticed that it was also a solution for the
Problem 29, a smaller not reported solution. Am I
right, Mr. Vantieghem? (Your
interpretation is perfectly correct ! Sincere greetings !,
June 2010)
On his turn the Problem 52 was taken by me from one entry
of the pages "Prime Curios!" from my friend G. L. Honaker, Jr.
So the circle is happily closed now, reporting very
surprising news:
1) Problem 29 & Problem 52 are two sides of a unique
problem: every solution of P29 is a solution of P52 and viceversa.
2) If this is so, the first Brier number produced was
not the 41 digits
SKB(28/9/98,
E. Brier) but
the 26 digits
SKB(26/5/2010, E. Vantieghem)
that was really discovered by Fred Cohen & J. L. Selfridge in 1975. 23
years before.
3) The current smallest known solution up today (Jun
2010) to both Problems 29 & 52 is the 24 digits
SKB(2007, Filaseta et al).
***
Accordingly this is the current history & status:
Status Table 
date 
Author 
Brier Number 
Digits 
28sep98 
Eric Brier 
29364695660123543278115025405114452910889 
41 
15ene00 
Yves Gallot 
623506356601958507977841221247 
30 
16ene00 
Yves Gallot 
3872639446526560168555701047 
28 
17ene00 
Yves Gallot 
878503122374924101526292469 
27 
1975 
F. Cohen & J.L.Selfridge 
47867742232066880047611079 
26 
26may10 
E. Vantieghem 
47867742232066880047611079 
26 
2007 
Filaseta et al 
143665583045350793098657 
24 
Next? 
Next? 
Next? 
Next? 
***
On June 20, Eric Brier wrote:
It is a strange feeling to discover that such numbers
were known in 1975!
Thank you for the information anyway.
I have been very surprised to read the construction
of Izotov to build Sierpinski numbers without covering sets.
It confirms that it is a domain where most probably
many things are still to be discovered.
***
On June 21, 2010 A. Wesolowski wrote:
Here are a few such numbers (26 digits each)
21867705038000924683065281
24155005016816795415535763
30902663634162389353963691
***
On Jan 3, 2014 I received an email from Christope Clavier
announcing that he has gotten 14 smaller Eric Brier numbers
Please find attached a list of fourteen Brier numbers that are all
smaller than the current record established by Filaseta et al. in 2007
according to your page.
Below each number I give the two covering sets related to k.2^n+1 and
k.2^n1 respectively.
I found all these numbers on the last day of 2013.
The smallest of them is the following 22digit Brier number :
3316923598096294713661
I will send an email to inform several people that I think may be
interested.
Nevertheless, do not hesitate to communicate by yourself to other people
if you wish.
43262598580503239091589 (22.6361)
{3, 5, 17, 257, 241, 97, 673}
{61, 41, 11, 31, 151, 331, 37, 13, 73, 19, 7, 3}
107711321583468432196343 (23.0323)
55465536577115049124007 (22.744)
{3, 5, 17, 13, 241, 97, 673}
{61, 41, 11, 31, 151, 331, 37, 109, 73, 19, 7, 3}
105404490005793363299729 (23.0229)
{61, 41, 11, 31, 151, 331, 37, 109, 73, 19, 7, 3}
{3, 5, 17, 13, 241, 97, 673}
3316923598096294713661 (21.5207)
{3, 5, 17, 13, 241, 97, 673}
{1321, 41, 11, 31, 151, 331, 37, 109, 73, 19, 7, 3}
32099522445515872473461 (22.5065)
92348240410439475192041 (22.9654)
50500982247079839441193 (22.7033)
28960674973436106391349 (22.4618)
{61, 41, 11, 31, 151, 331, 37, 109, 73, 19, 7, 3}
{3, 5, 17, 13, 241, 97, 257}
88595984169672153528691 (22.9474)
12607110588854501953787 (22.1006)
10439679896374780276373 (22.0187)
76719416286801468925067 (22.8849)
21444598169181578466233 (22.3313)
{3, 5, 17, 13, 241, 97, 257}
{61, 41, 11, 31, 151, 331, 37, 109, 73, 19, 7, 3}
Smallest Brier number of this file:
3316923598096294713661 (21.5207)
One day later I received the following note from Wilfrid
Keller
Dear Christophe Clavier,
I am very pleased to congratulate you on the discovery
(or rather: construction) of a new smallest known "Brier
number".
It's a particularly happy event, as the 23digit value
10439679896374780276373 (22.0187)
included in your list had just been presented in a paper
by Dan Ismailescu and Peter Seho Park
( https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf ,
submitted in June 2013, published as recently as
December 5, 2013).
So your 22digit example remains unaffected as the current
"record"  a remarkable achievement!
Thank you for letting me know your results as quickly.
Best regards and a Happy New Year!
Wilfrid Keller
Clavier gave me permission to announce his results to OEIS site and you may
see these here
A076335.
Pending to receive from
Christope a small note explaining his approach in order to get his
remarkable results (all
these numbers on the last day of 2013)...
Here is his explanation...
Before writing about the way I found the 14 Brier numbers smaller than
the 24digit record from Filatesa et al. I must say that Wilfrid Keller
informed me on Junuary 3, 2014 about a recent result from Dan Ismailescu
and Peter Seho Parkthat that I was unaware of. They wrote a paper (https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf)
that has been published in the Journal of Integer Sequences on December
5, 2013 where they provide a new smallest known Brier number:
10439679896374780276373 (23 digits).
This number is present in my list and is only beated by the smallest of
the list: 3316923598096294713661 (22 digits).
So this means I beated the current record only once and not fourteen
times as I thought!
Note also that 10439679896374780276373 is prime while
3316923598096294713661 is not so that Dan Ismailescu and Peter Seho
Parkthat still hold the smallest known *prime* Brier number.
Regarding the way I found my numbers, there is no much to say. I merely
tried to find (in a careful but non systematic way) a good pair of
covering sets. I would define what I mean by a good pair of covering
sets by the following efficiency metric: \mu = \pi / N where \pi is
the product of the primes belonging to the two sets divided by 3
(because the prime 3 appears twice and must be counted once) and where N
is the number of odd Brier numbers produced by the CRT modulo \pi. The
rational behind this metric is that if we modelize the distribution of
these N odd integers as being uniform in the interval [0,\pi1], then we
must expect that the smallest one is about as small as \pi / N. The
magnitude of the smallest Brier number produced by some pair of covering
sets is thus influenced by \pi and N which are both related to the pair
of sets, as well as by an independant luck/unluck factor. As I will
expose shortly, the 22digit Brier number I found is mainly due to luck!
Four pairs of covering sets are involved in my list of 14 Brier numbers:
(note: the size (log in base 10) of each Brier number is given beside it
into parenthesis)
pair 1: {3, 5, 17, 257, 241, 97, 673} and {61, 41, 11, 31, 151, 331, 37,
13, 73, 19, 7, 3} or viceversa which produced one number:
 43262598580503239091589 (22.6361)
pair 2: {3, 5, 17, 13, 241, 97, 673} and {61, 41, 11, 31, 151, 331, 37,
109, 73, 19, 7, 3} or viceversa which produced three numbers:
 107711321583468432196343 (23.0323)
 55465536577115049124007 (22.744)
 105404490005793363299729 (23.0229)
pair 3: {3, 5, 17, 13, 241, 97, 673} and {1321, 41, 11, 31, 151, 331,
37, 109, 73, 19, 7, 3} or viceversa which produced only the record
number:
 3316923598096294713661 (21.5207)
pair 4: {3, 5, 17, 13, 241, 97, 257} and {61, 41, 11, 31, 151, 331, 37,
109, 73, 19, 7, 3} or viceversa which produced as much as nine numbers:
 32099522445515872473461 (22.5065)
 92348240410439475192041 (22.9654)
 50500982247079839441193 (22.7033)
 28960674973436106391349 (22.4618)
 88595984169672153528691 (22.9474)
 12607110588854501953787 (22.1006)
 10439679896374780276373 (22.0187)
 76719416286801468925067 (22.8849)
 21444598169181578466233 (22.3313)
Note that for each of these pairs of covering sets N is equal to
3,317,760 so that the efficiency figures for each pair are respectively:
pair 1 => \mu = 22.7914
pair 2 => \mu = 22.4189
pair 3 => \mu = 23.7545
pair 4 => \mu = 22.0008
One can notice that the last pair have the better efficiency figure
which is reflected by the fact that it produced as much as nine small
Brier numbers. Also note that its smallest number has a size (22.0187)
which is quite close to the expected one (22.0008). On the contrary it
is curious that the record number was produced by the third pair which
was a priori the less promising one (because of the seemingly bad ideas
of having replaced 257 by 673 and 61 by 1321) . One can define a kind of
(un)luck index \delta as the discrepancy between the expected and the
actual sizes of the smallest number produced by a pair: \delta = \mu 
log_10(smallest number). The more \delta is, the more luck we had, and
conversely.
With this notion in hand we can list the luck index of the smallest
number of each pair:
pair 1 => \delta = 22.7914  22.6361 = +0.1553 (slightly lucky)
pair 2 => \delta = 22.4189  22.744 = 0.3251 (somewhat unlucky)
pair 3 => \delta = 23.7545  21.5207 = +2.2338 (exceptionnaly lucky)
pair 4 => \delta = 22.0008  22.0187 = 0.0179 (near as expected)
As one can see, finding this precise record number in this precise pair
of covering set was a quite lucky event. It seems to me that for large
\delta values one can see 1/10^{\delta} as an approximation of the
probability that the pair of covering sets would produce such a small
number. In that particular case this gives a probability close to 1 over
170.
One can ask whether a smaller Brier number constructed from covering
sets may exist. Though the answer is uncertain, the analysis above tend
to inspire pessimism. Indeed it is probable that the most promising
pairs have already been considered either by previous small Brier
numbers seekers or by myself. Thus a winning pair would probaly have its
\mu value greater than 23.7545 and consequently should give occurence to
a luck index probably much greater than +2.2338 which seems improbable.
Of course this reasonning is definitely not a proof of the non existence
of smaller Brier numbers.
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