Problems & Puzzles: Problems Problem 29.- Brier Numbers A Sierpinski Number is an integer k such that k.2n+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.2n-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.2n+1 and k.2n-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:
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) Solution 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:
So, now the smallest known Brier number to beat is: SKB(15/1/2000, Y. Gallot) = 623506356601958507977841221247 (30 digits) Minutes after receiving his e-mail 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: 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. 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 semi-automatic) was successful. I used no special rules but a brute-force 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 re-discovered 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 10-11 digits will be confirmed by the method used by Brier and Gallot... *** Ink was still fresh when the following message from Gallot arrived: { 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:
*** 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 Brier-alpha 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. ***
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