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Old 2008-12-17, 09:24   #3
gd_barnes
 
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May 2007
Kansas; USA

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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
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