View Single Post
 2008-12-17, 09:24 #3 gd_barnes     May 2007 Kansas; USA 27BE16 Posts Not only can we generalize the algebraic factors for a specific factor across bases, I believe we can generalize all prime "numeric" factors across all Riesel bases that combine with algebraic factors to make full covering sets. Analysis: 1. See the above 2 posts for factors of 5 and 37. 2. The same type of scenario occurs for a factor of 13 on base 12 (and likely bases 25, 38, 51, etc. if they are applicable), a factor of 29 on base 28 (and likely bases 57, 86, 115, etc. if they are applicable), a factor of 17 on base 33 (and likely bases 50, 67, 84, etc. if they are applicable), and a factor of 61 on base 60 (and likely bases 121, 182, 243, etc. if they are applicable). See the web pages. Therefore I propose the following conjecture to the math community for Riesel bases: In addition to having full algebraic factors on k's and bases that are perfect squares, there are k's that are perfect squares for many bases that are NOT perfect squares that have a numeric prime factor (f) that combines with algebraic factors to make a full covering set in the following scenarios: For any prime factor f and any base b, I conjecture that for the following set of conditions: b==(f-1 mod f) -and- f==(1 mod 4) -and- k=m^2 -and- m==(x or y mod f) -and- x+y = f -and- x and y are unique for each f -and- 0 < x, y < f That f is a prime factor on odd-n and algebraic factors of the form [m*b^(n/2)-1]*[m*b^(n/2)+1] are present on even-n and that these combine to make a full covering set for the form k*b^n-1. Listing for all factors up to 1035: Code: bases b== factor x & y-values 4mod5 5 2, 3 (2, 3, & 5 are Fibonacci numbers) 12mod13 13 5, 8 (5, 8, & 13 are Fibonacci numbers) 16mod17 17 4, 13 28mod29 29 12, 17 36mod37 37 6, 31 40mod41 41 9, 32 52mod53 53 23, 30 60mod61 61 11, 50 72mod73 73 27, 46 88mod89 89 34, 55 (34, 55, & 89 are Fibonacci numbers) 96mod97 97 22, 75 100mod101 101 10, 91 108mod109 109 33, 76 112mod113 113 15, 98 136mod137 137 37, 100 148mod149 149 44, 105 156mod157 157 28, 129 172mod173 173 80, 93 180mod181 181 19, 162 192mod193 193 81, 112 196mod197 197 14, 183 228mod229 229 107, 122 232mod233 233 89, 144 (89, 144, & 233 are Fibonacci numbers) 240mod241 241 64, 177 256mod257 257 16, 241 268mod269 269 82, 187 276mod277 277 60, 217 280mod281 281 53, 228 292mod293 293 138, 155 312mod313 313 25, 288 316mod317 317 114, 203 336mod337 337 148, 189 348mod349 349 136, 213 352mod353 353 42, 311 372mod373 373 104, 269 388mod389 389 115, 274 396mod397 397 63, 334 400mod401 401 20, 381 408mod409 409 143, 266 420mod421 421 29, 392 432mod433 433 179, 254 448mod449 449 67, 382 456mod457 457 109, 348 460mod461 461 48, 413 508mod509 509 208, 301 520mod521 521 235, 286 540mod541 541 52, 489 556mod557 557 118, 439 568mod569 569 86, 483 576mod577 577 24, 553 592mod593 593 77, 516 600mod601 601 125, 476 612mod613 613 35, 578 616mod617 617 194, 423 640mod641 641 154, 487 652mod653 653 149, 504 660mod661 661 106, 555 672mod673 673 58, 615 676mod677 677 26, 651 700mod701 701 135, 566 708mod709 709 96, 613 732mod733 733 353, 380 756mod757 757 87, 670 760mod761 761 39, 722 768mod769 769 62, 707 772mod773 773 317, 456 796mod797 797 215, 582 808mod809 809 318, 491 820mod821 821 295, 526 828mod829 829 246, 583 852mod853 853 333, 520 856mod857 857 207, 650 876mod877 877 151, 726 880mod881 881 387, 494 928mod929 929 324, 605 936mod937 937 196, 741 940mod941 941 97, 844 952mod953 953 442, 511 976mod977 977 252, 725 996mod997 997 161, 836 1008mod1009 1009 469, 540 1012mod1013 1013 45, 968 1020mod1021 1021 374, 647 1032mod1033 1033 355, 678 This is exciting stuff for making the Riesel conjectures easier in the future. Thanks, Gary Last fiddled with by gd_barnes on 2010-10-08 at 17:09 Reason: add all factors up to 1024