I.- The Basis of the Search

OK, Folks of the P/Ba  ("Primes through the Brier-Alfa" team)

The idea is to continue the search that up today has been carried only and working alone by Wilfrid Keller, related to the search for Brier-Alpha number (please read the Problems 29 & 30).

I have been informed by Keller that his search around the k = 1355477231 is now  near the exponent n = 210000+ & 260000- ( this means two future fairly good primes if his search for the first Brier number fails).

Keller also told me that he will no look far beyond k>2^31 because this is the limit of Proth and because he has already 25 "good" candidates to the Brier-Alpha (or for 50 couples of primes...)

Hola Carlos:

Answering your question, I would like to inform you that I intended to continue the search for possible
"Brier candidates" up to  k < 2^31,  just the limit tractable with Yves' program.  Actually, the first
undecided value  k > 2^31  is  k = 2152690373,  with no exponent  n < 6000  giving a prime at either side.
So, as from my part, anyone is invited to take up the search at that point.

Below  2^31,  I am left with about 25 cases.  Though I had to neglect a little this project for some time
because I have been heavily involved in a completely different number theoretical investigation, I would
very much appreciate it if that range could be kept reserved for me until it is "finished".

During the computations there seemed to have appeared one more "suspicious" case, but it finally produced the prime  1501846421*2^125703 + 1  (no prime  k.2^n - 1 up to that limit).  The size of that prime and the some- how "similar behavior" of both sequences, as compared to those for  k = 1355477231  lets me hope that this  k will also "fall" through some very large prime (as you have been predicting).  The limits reached so far are n <= 210000  for +1 and  n <= 260000  for -1.

Best wishes, and good luck in your supplementary search. Saludos, Wilfrid

Then we have the road free for k>2152690373. For k values larger than 2^31 the only extra-cost we are going to pay is that - being the Proth out of the game - we will have to switch to PRIMEFORM (by Chris Nash) in the "file mode" to test the primality for those promising k values previously detected by a sieving code ( I suggest Ubasic). A promising k value is such that k*2^n+1 and k*2^n-1 are composite up to certain n value (6000?).

This kind of search of primes - through a highly composite producer k value - is of the same nature than the organized around the Sierpinski numbers. As probably you know, the k value  4847 (see http://www.prothsearch.net/sierp.html ) right now has to Payam Semidoost at the door entrance of a very very high prime number (n>800,000) that will constitute by itself a Proth prime record ready to enter to the Top
Ten list.

II.- Structure of the search

a) Define certain range for k>2^2152690373 to work the search of Brier -Alpha numbers or the first two primes to appear.
b) Inside that range, detect all the promising k values using a code in Ubasic that I have already made.
c) Run the code pfgw trying to see if the compositeness of k*2^n+/-1 remains for n values higher & higher.

Each one of us will work over certain predefined k-ranges in both operations (Ubasic/pfgw) reporting each promising k value when detected and later the two n values where this k provided the first two primes.

Whenever a certain k value climbs up to high n values (let's say 50,000 or 100,000) without providing any prime, we may decide to join more than one person at a time in this very promissory k value.

I will have a "secret" page (this is that page) in my site to keep tracking our collective progress, obviously accessible to all of us, the members of the team.

Our internal agreement could be that we will only collectively publish the primes gotten and above certain value on n (for example n>200,000 or other suggested bound value)

III.- The General Range and the sub-ranges proposed

The general range proposed is simply the contiguous upper and of the same size than the one examined by Keller during a year or so alone (3 - 2152690373) subdivided between each member of the team. The idea is that if he got in a similar range one very good & promising k value and 25 other expecting more investigation, we should have similar results but earlier.

Ranges proposed & status of search.

Other Comments

Hunting primes via Brier numbers has one "danger": to hunt one real Breir number or the Brier-Alpha number. You must realize that this is a pseudo-danger, of the same nature than the "danger" that a miner can have for discovering oil while he was mining for gold... (question: Is Payam working useless over a "Sierpinski" number?)

C. Rivera
March 7, 2001

The Ubasic Sieve step

This is the Ubasic Code suggested to sieve k values such that k*2^n+1 and k*2^n-1 are composite for all 1=>n=>3200.

10 K=2152690375:NM=3200
15 open "brierk.txt" for append as #1
20 print "Candidates to Brier-Alpha numbers: k*2^n+/-1 composites for all n=>1"
30 for N=1 to NM
40 Q=K*2^N+1:gosub *SPSP:if PRIMO=1 then cancel for:goto 80
50 Q=K*2^N-1:gosub *SPSP:if PRIMO=1 then cancel for:goto 80
60 next N
70 print:print K,"Composite up to n = ";NM
72 print #1:print #1,K,"Compuesto hasta n=";NM,time
80 K=K+2:if (K-1)@1000000=0 then print K,:print #1,K,
90 goto 30
100 *SPSP:PRIMO=0:PDQ=prmdiv(Q)
110 if Q=PDQ then PRIMO=1:goto 250
120 if and{1<PDQ,PDQ<Q} then goto 250
130 B=irnd@11998+2
140 T=Q-1:A=0
150 for II=1 to 10000
160 if even(T)=0 then cancel for:goto 190
170 T=T\2:A=A+1
180 next II
190 W=modpow(B,T,Q)
200 if or{W=1,W=Q-1} then PRIMO=1:goto 250
210 for II=1 to A-1
220 W=(W*W)@Q
230 if W=Q-1 then PRIMO=1:cancel for:goto 250
240 next II
250 return

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The second step using pfgw

The second step definitively is easier to do using the code pfgw, the following way

1. Download pfgw from here (local source, debugged code sent by Michael Bell) and unzip it a a separate folder (let's suppose the folder is C:\pfgw)

2. Create the "name".txt file with the following two lines (let's suppose that name = kbriera)

ABC2 3022075061*2^$a+1|3022075061*2^$a-1
a: from 3200 to 6000

(this will test the k value 3022075061 for a probable primality test with the two expressions k*2^n+1 and k*2^n-1 for n=3200 to 6000)

3. Go to the DOS window and change the prompt to C:\pfgw 

4. Type in the following command line: pfgw kbrier.txt -f

At the end of the run verify if you have a prime in the automatically created pfgw.log file.

Testing several k values for the same range.

Let's suppose that you want to test 3 k values (303, 3001, 6733) for the range n =12000 to 15000. Then type in the kbriera.txt file the following 3 lines

ABC2 $a*2^$b+1 | $a*2^$b-1
a: in { 303 3001 6733 }     <--  no commas just spaces between brackets & numbers and between numbers
b: from 12000 to 15000

March 11, 2001

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Thinking aloud

What a strange way of getting a large prime k*2^n+1 or k*2^n-1  we are attempting...

We are trying to find a k that produces no primes for a large range of n but at the same time desiring that if this prime can not be avoided then it becomes when n is large.

Does this has sense?

Is not a better strategy to detect a k that produces a large quantity of primes for certain low range of n (for example for n<3200) and the simply switching to search directly in large values of n?

What is a better strategy?

As a matter of fact there are people working under both strategies.

People like us (Brier project) are these people that are around the Sierpinski and the Riesel projects both maintained by Keller & Ballinger. Certainly we are not alone in this strategy. The people that is using the other strategy are not working in team (as far as I know) but certainly are these that uses sieving tools that produces a "weight" (the quantity of primes in certain range of n) for the k chosen.

In the very beginning of my career as prime hunter I even choose another strategy: I simply selected a n value fixed ( a beauty n number according to my personal taste) and then simply I started running over all the odd values of k (using the Proth.exe of course). As a matter of fact for this way of proceeding Paul Jobling has a sieve program, the named NewPGen.

I wonder if somebody has analyzed the virtues of these different  approaches to getting large primes...

March 15, 2001
CR

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What to do when we detect a large enough probable prime (PRP) using the pfgw before reporting to the Chris Caldwell Primes list? How to run the rigorous primality test in the pfgw?

This is the Michael Bell's answer:

After you have found a PRP you must then run a N-1 or N+1 test as appropriate, eg.

pfgw -q3204780493*2^3395-1 -tp
pfgw -q3204780493*2^5580+1 -tm

-tp tells pfgw to do an N+1 test, -tm tells it to do a N-1 test. If you have more than one PRP to prove then you can make a file containing a list of numbers to test just as in normal operation.

26/03/2001

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Michael Bell has sent two sieving codes (basieve1 and basieve2) particularly useful when in our working k range we are left with only one k survivor.

basieve1 & basieve2 will ask for the k survivor value and the extreme values of the n range we want to sieve (that is to say, to discard n values such that or k*2^n+1 or k*2^n-1 are composite). It will produce a file output.txt with the remaining n values for each expression, that will be now the input file for the pfgw step for that n range.

The second code, basieve2  will only be helpful if the k survivor has produced a prime let's say with the k*2^n+1 expression and we want to continue the search for the prime with the alternative expression k*2^n-1.

28/4/01

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Just to have an idea where our first prime will be ranked if we got Bingo! in any of the working ranges please click here: the Top 100 list in the Caldwell's database.

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