The Riesel-Problem

Started: Aug 26, 2007
Last update: Nov 24, 2007

Latest news: On Oct 28, 2007 a new Riesel Prime for k=469949 was found during a double check!

Special thanks to Wilfrid Keller for the data. See also his page here.

Definition :

Theorem: There exist infinitely many odd integers k such that k * 2n-1 is composite for every n > 1.

Hans Riesel proved this in 1956 and showed that k0 = 509203 has got this property and also all multipliers kn = k0 + 11184810 * n (n > 1).
Such numbers are called Riesel numbers. The Riesel problem is finding the smallest Riesel number.

Conjecture: k = 509203 is the smallest Riesel number.

To prove this conjecture, for every k < 509203 a prime k * 2n-1 must be found. It's sufficient to find the first exponent n giving a prime.
So the search is seperated in stages fm where the first prime exponent is in the interval 2m < n < 2m+1.

Data :

Number of k's with first prime in the interval 2m < n < 2m+1. (green: all these k's are listed in the summary pages)

m 2m 2m+1 # k's contributors(primes found) Date completed
0 1 2 39867 Keller, Gallot 1992
1 2 4 59460 Keller, Gallot 1992
2 4 8 62311 Keller, Gallot 1992
3 8 16 45177 Keller, Gallot 1992
4 16 32 24478 Keller, Gallot 1992
5 32 64 11668 Keller, Gallot 1992
6 64 128 5360 Keller, Gallot 1992
7 128 256 2728 Keller, Gallot 1992
8 256 512 1337 Keller, Gallot 1992
9 512 1024 785 Keller, Gallot 1992
10 1024 2048 467 Keller, Gallot 1992
11 2048 4096 289 Gallot ?
12 4096 8192 191 Gallot ?
13 8192 16384 125 Gallot 11.05.1998
14 16384 32768 87 Ballinger(21), Keller(25), Key(41) 14.11.1998
15 32768 65536 62 Ballinger(5), Brazier(1), Caldwell(2), Keller(13), Key(22), Kuechler(1), Linton(18) 09.04.1999
16 65536 131072 38 Ballinger(2), Gallot(1), Keller(4), Key(5), Linton(25), Zeisel(1) 22.10.1999
17 131072 262144 35 Aitsen(1), Ballinger(2), Baxter(1), Gallot(1), Haeberle(1), Heylen(1), Linton(25), Pirson(1), Szmidt(1), Zeisel(1) 13.07.2001
18 262144 524288 25 Andrews(1), Ballinger(4), Davis(1), Haeberle(1), Heylen(2), Keiser(1), Kuechler(1), Kuosa(1), Linton(6), Pirson(1), Rodenkirch(1), Schmid(1), Szmidt(2), Wolfe(1), Zeisel(1) 04.12.2003
19 524288 1048576 21 Haeberle(10), Ballinger(1), Heylen(1), Linton(1), Schmid(1), RieselSieve(7) 23.09.2004
20 1048576 2097152 17 Linton(1), Ballinger(1), RieselSieve(15) 28.10.2007
21 2097152 4194304 >6? RieselSieve(6) ?
? >4194304 Infinity 66 remain see RieselSieveProject  


f(13) = 125 where k*2n-1 with 8192 < n < 16384

kn kn  kn kn  kn kn  kn kn
86818458 1345110562 160338547 195079713 249598976 2981913052 3676112018 435439419
446698560 4643915088 469019346 5022711893 5072914788 508018970 527778333 5875311955
5947310187 6421313839 6551911148 7623713085 8593314383 864378933 9443912424 9458913436
963679409 977239447 1088179429 11515110930 1228798812 12638915576 13221715281 13642113534
1391838419 15610110798 15795716065 16492113258 16549114486 16648316243 1689079225 17512113402
18352315387 18605914932 19258111006 19452711573 19481915256 1958759350 2043978673 2057818334
2073118998 2096478561 21232113838 21355312771 21749911244 22082915960 2209318930 22568910800
2285099980 23689314751 23765915856 23988713473 24198711277 24556310319 24673112078 2475979917
24804711613 25009115690 25350710833 26026116202 2636699864 2664598852 28239511218 2834819394
2889439227 2907078457 29146310791 29364711225 29530311711 2999098828 30008915160 30126712493
30368910916 30537711457 30663710273 3076318254 31084310775 31286916116 31796914300 3228778328
32557110610 3260478801 33535314591 34596710269 3480379113 35212314995 3533759382 35569310203
3631199712 3633599788 3640799052 3691639099 3735739167 37509712421 3796939655 40009312815
40196914732 41437310703 41555911444 41793714725 4202399488 42463113166 43043915600 43409910420
43526916060 43700910196 44914313347 45721312839 4597018706 4639619794 47154715141 48271115046
48936711393 4910179141 50120910008 5029799044 50761315019


f(14) = 87 where k*2n-1 with 16384 < n < 32768

kn kn  kn kn  kn kn  kn kn
121125242 1039131914 1045119246 1434725997 2288916692 2483920436 2582526961 3463125390
6346317219 6387721409 6905931344 6930132437 7927321107 8591928516 9059521643 9541128454
9791920872 11422322463 11591516389 12570719021 12911919584 13122730441 13132116910 14261918716
14815125790 14946722949 16498117806 16970918856 17160721117 17707326147 18013319879 18403123906
19097923244 19456324343 20783929140 21865130314 22744124590 22807127662 23001720045 23322716517
23637719693 24484117890 26239119766 27871317159 28189129950 30061121194 30204719737 30674918916
31377716713 31543323183 31846317707 32050326627 32357930324 32373718001 32868725357 32941316947
34307922092 34720127574 35017325523 35061731493 35399916932 35992322463 36081718285 36754317831
36846729221 37099118522 37231128974 37630318335 38479122598 38722320951 39167917432 39767927540
40353727505 40642318155 41080331895 41972919632 43510120382 43981120290 44736722149 45019323059
45695920016 46296724757 48390718089 48424122158 48667727910 50564918644 50897917088


f(15) = 62 where k*2n-1 with 32768 < n < 65536

kn kn  kn kn  kn kn  kn kn
381740381 1682957728 2507934660 3356952512 4410765173 5731140498 5735962500 6438740249
8131364619 8298749489 8683344747 10454939744 10706943280 11188354415 11653159782 13321134342
13795133094 14092133234 15272948348 16478948268 17537937676 17747941032 20922743481 21371954996
22051154890 23383332983 25458742953 26440335147 27807756913 28890760485 28940937444 29758162982
30038934768 30063760109 30835145326 31420338267 31843140618 32053146318 33832149538 34144342575
34805934660 35355751833 35824340707 38119343627 38222939004 38562734229 40187939864 40452737265
40625340948 41048947596 42340334691 42449934168 43117363643 43807155986 45343349725 45358158266
46186161762 46508933868 46831159978 47883755081 48781142478 49764759825


f(16) = 38 where k*2n-1 with 65536 < n < 131072

kn kn kn  kn kn  kn kn
3068978560 4198372347 48703109415 7709969356 8332383079 9556176754 103811111043
12443994176 12466368939 13162369099 13809786401 150103111391 16294374319 17006996828
17974399731 182939127180 21796395111 22489192214 24553384667 27537789937 296321101594
30115178326 30211196282 332201103498 358889122900 362707119925 36433192626 39231774281
39292370827 39928167798 40442985344 420067125733 444847112497 462079105187 46490973740
46763974496 477311105214 483479115376


f(17) = 35 where k*2n-1 with 131072 < n < 262144

kn kn kn  kn kn  kn kn
11519164444 14459171144 25229238652 37837180783 81517258321 105569235200 105697223233
107167159161 111253165379 111763155551 130297136645 132071202098 132599206032 144817258857
159821168770 178747144789 185767149009 190229141576 217807243537 256267148941 281143187639
285191201138 307211241978 321043238303 325859156148 331139201240 370421201442 392737248517
393209221216 408247205469 438523135415 466783245839 471127157629 485773216487 499031139894


f(18) = 25 where k*2n-1 with 262144 < n < 524288

kn kn kn  kn kn  kn kn
27253272347 39269287048 42779322908 43541507098 46271428210 104917340181 130139280296
144643498079 148901360338 159371284166 189463324103 201193457615 220063306335 235601295338
245051285750 267763264115 277153429819 299617428917 376993293603 382691431722 398533419107
401617470149 416413424791 443857369457 465869497596


f(19) = 21 where k*2n-1 with 524288 < n < 1048576

kn kn kn  kn kn kn  kn
659800516 89707578313 93997864401 98939575144 103259615076 109897630221 126667626497
170591866870 204223696891 212893730387 215503649891 220033719731 222997613153 246299752600
261221689422 279703616235 309817901173 357491609338 401143532927 458743547791 460139779536


f(20) = 18 where k*2n-1 with 1048576 < n < 2097152

kn kn kn  kn kn kn  kn
710091185112 1104131591999 1497971414137 1508471076441 1527131154707 1920891395688 2348471535589
3256271472117 3450671876573 3501071144101 3576591779748 4127171084409 4176431800787 4679171993429
4699491649228 5006211138518 5025411199930 5046131136459


f(21) >= 6 ? where k*2n-1 with 2097152 < n < 4194304

kn kn kn  kn kn kn
267732465343 1144872198389 1965972178109 2752932335007 3426732639439 4504572307905


66 remaining k's with no prime for n < 2.8 M

k k k  k k k  k k k  k k k
2293 9221 2366931859 3847340597 4666365531 6711774699 8104193839
97139 107347 113983121889 123547129007 141941143047 146561161669 162941191249
192971206039 206231215443 226153234343 245561250027 252191273809 304207315929
319511324011 325123327671 336839342847 344759353159 362609363343 364903365159
368411371893 384539386801 397027398023 402539409753 415267428639 444637470173
474491477583 485557485767 494743502573