 mersenneforum.org A Sierpinski/Riesel-like problem
 Register FAQ Search Today's Posts Mark Forums Read  2020-07-08, 17:08 #870 sweety439   Nov 2016 3·5·137 Posts All n must be >= 1. All MOB k-values need an n>=1 prime (unless they have a covering set of primes, or make a full covering set with all or partial algebraic factors) MOB k-values such that (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not prime are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b. There are some MOB k-values such that (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is prime which do not have an easy prime for n>=1: GFN's and half GFN's: (all with no known primes) S2 k=65536 S3 k=3433683820292512484657849089281 S4 k=65536 S5 k=625 S6 k=1296 S7 k=2401 S8 k=256 and k=65536 S9 k=3433683820292512484657849089281 S10 k=100 S11 k=14641 S12 k=12 Other k's: S2 k=55816: first prime at n=14536 S2 k=90646: no prime with n<=6.6M S2 k=101746: no prime with n<=6.6M S3 k=621: first prime at n=20820 S4 k=176: first prime at n=228 S5 k=40: first prime at n=1036 S6 k=90546 S7 k=21: first prime at n=124 S9 k=1746: first prime at n=1320 S9 k=2007: first prime at n=3942 S10 k=640: first prime at n=120 S24 k=17496 S155 k=310 S333 k=1998 R2 k=74: first prime at n=2552 R2 k=674: first prime at n=11676 R2 k=1094: first prime at n=652 R4 k=19464: no prime with n<=3.3M R6 k=103536: first prime at n=6474 R6 k=106056: first prime at n=3038 R10 k=450: first prime at n=11958 R10 k=16750: no prime with n<=200K R11 k=308: first prime at n=444 R18 k=324: first prime at n=25665 R21 k=84: first prime at n=88 R23 k=230: first prime at n=6228 R27 k=594: first prime at n=36624 R40 k=520: no prime with n<=1K R42 k=1764: first prime at n=1317 R48 k=384: no prime with n<=200K R66 k=1056: no prime with n<=1K R78 k=7800: no prime with n<=1K R88 k=3168: first prime at n=205764 R96 k=9216: first prime at n=3341 R210 k=44100: first prime at n=19817 R306 k=93636: first prime at n=26405 R396 k=156816: no prime with n<=50K R591 k=1182: first prime at n=1190 R954 k=1908: first prime at n=1476 R976 k=1952: first prime at n=1924 R1102 k=2204: first prime at n=52176 R1297 k=2594: first prime at n=19839 R1360 k=2720: first prime at n=74688 Last fiddled with by sweety439 on 2020-07-08 at 17:21   2020-07-08, 17:58   #871
sweety439

Nov 2016

80716 Posts Quote:
 Originally Posted by sweety439 In Riesel conjectures, if k=m^2 and m and b satisfy at least one of these conditions, then this k should be excluded from the Riesel base b problem, since it has algebraic factors for even n and it has a single prime factor for odd n, thus proven composite by partial algebraic factors list all such mod <= 2048 Code: m b = 2 or 3 mod 5 = 4 mod 5 = 5 or 8 mod 13 = 12 mod 13 = 3 or 5 mod 8 = 9 mod 16 = 4 or 13 mod 17 = 16 mod 17 = 12 or 17 mod 29 = 28 mod 29 = 7 or 9 mod 16 = 17 mod 32 = 6 or 31 mod 37 = 36 mod 37 = 9 or 32 mod 41 = 40 mod 41 = 23 or 30 mod 53 = 52 mod 53 = 11 or 50 mod 61 = 60 mod 61 = 15 or 17 mod 32 = 33 mod 64 = 27 or 46 mod 73 = 72 mod 73 = 34 or 55 mod 89 = 88 mod 89 = 22 or 75 mod 97 = 96 mod 97 = 10 or 91 mod 101 = 100 mod 101 = 33 or 76 mod 109 = 108 mod 109 = 15 or 98 mod 113 = 112 mod 113 = 31 or 33 mod 64 = 65 mod 128 = 37 or 100 mod 137 = 136 mod 137 = 44 or 105 mod 149 = 148 mod 149 = 28 or 129 mod 157 = 156 mod 157 = 80 or 93 mod 173 = 172 mod 173 = 19 or 162 mod 181 = 180 mod 181 = 81 or 112 mod 193 = 192 mod 193 = 14 or 183 mod 197 = 196 mod 197 = 107 or 122 mod 229 = 228 mod 229 = 89 or 144 mod 233 = 232 mod 233 = 64 or 177 mod 241 = 240 mod 241 = 63 or 65 mod 128 = 129 mod 256 = 16 or 241 mod 257 = 256 mod 257 = 82 or 187 mod 269 = 268 mod 269 = 60 or 217 mod 277 = 276 mod 277 = 53 or 228 mod 281 = 280 mod 281 = 138 or 155 mod 293 = 292 mod 293 = 25 or 288 mod 313 = 312 mod 313 = 114 or 203 mod 317 = 316 mod 317 = 148 or 189 mod 337 = 336 mod 337 = 136 or 213 mod 349 = 348 mod 349 = 42 or 311 mod 353 = 352 mod 353 = 104 or 269 mod 373 = 372 mod 373 = 115 or 274 mod 389 = 388 mod 389 = 63 or 334 mod 397 = 396 mod 397 = 20 or 381 mod 401 = 400 mod 401 = 143 or 266 mod 409 = 408 mod 409 = 29 or 392 mod 421 = 420 mod 421 = 179 or 254 mod 433 = 432 mod 433 = 67 or 382 mod 449 = 448 mod 449 = 109 or 348 mod 457 = 456 mod 457 = 48 or 413 mod 461 = 460 mod 461 = 208 or 301 mod 509 = 508 mod 509 = 127 or 129 mod 256 = 257 mod 512 = 235 or 286 mod 521 = 520 mod 521 = 52 or 489 mod 541 = 540 mod 541 = 118 or 439 mod 557 = 556 mod 557 = 86 or 483 mod 569 = 568 mod 569 = 24 or 553 mod 577 = 576 mod 577 = 77 or 516 mod 593 = 592 mod 593 = 125 or 476 mod 601 = 600 mod 601 = 35 or 578 mod 613 = 612 mod 613 = 194 or 423 mod 617 = 616 mod 617 = 154 or 487 mod 641 = 640 mod 641 = 149 or 504 mod 653 = 652 mod 653 = 106 or 555 mod 661 = 660 mod 661 = 58 or 615 mod 673 = 672 mod 673 = 26 or 651 mod 677 = 676 mod 677 = 135 or 566 mod 701 = 700 mod 701 = 96 or 613 mod 709 = 708 mod 709 = 353 or 380 mod 733 = 732 mod 733 = 87 or 670 mod 757 = 756 mod 757 = 39 or 722 mod 761 = 760 mod 761 = 62 or 707 mod 769 = 768 mod 769 = 317 or 456 mod 773 = 772 mod 773 = 215 or 582 mod 797 = 796 mod 797 = 318 or 491 mod 809 = 808 mod 809 = 295 or 526 mod 821 = 820 mod 821 = 246 or 583 mod 829 = 828 mod 829 = 333 or 520 mod 853 = 852 mod 853 = 207 or 650 mod 857 = 856 mod 857 = 151 or 726 mod 877 = 876 mod 877 = 387 or 494 mod 881 = 880 mod 881 = 324 or 605 mod 929 = 928 mod 929 = 196 or 741 mod 937 = 936 mod 937 = 97 or 844 mod 941 = 940 mod 941 = 442 or 511 mod 953 = 952 mod 953 = 252 or 725 mod 977 = 976 mod 977 = 161 or 836 mod 997 = 996 mod 997 = 469 or 540 mod 1009 = 1008 mod 1009 = 45 or 968 mod 1013 = 1012 mod 1013 = 374 or 647 mod 1021 = 1020 mod 1021 = 255 or 257 mod 512 = 513 mod 1024 = 355 or 678 mod 1033 = 1032 mod 1033 = 426 or 623 mod 1049 = 1048 mod 1049 = 103 or 958 mod 1061 = 1060 mod 1061 = 249 or 820 mod 1069 = 1068 mod 1069 = 530 or 563 mod 1093 = 1092 mod 1093 = 341 or 756 mod 1097 = 1096 mod 1097 = 354 or 755 mod 1109 = 1108 mod 1109 = 214 or 903 mod 1117 = 1116 mod 1117 = 168 or 961 mod 1129 = 1128 mod 1129 = 140 or 1013 mod 1153 = 1152 mod 1153 = 243 or 938 mod 1181 = 1180 mod 1181 = 186 or 1007 mod 1193 = 1192 mod 1193 = 49 or 1152 mod 1201 = 1200 mod 1201 = 495 or 718 mod 1213 = 1212 mod 1213 = 78 or 1139 mod 1217 = 1216 mod 1217 = 597 or 632 mod 1229 = 1228 mod 1229 = 546 or 691 mod 1237 = 1236 mod 1237 = 585 or 664 mod 1249 = 1248 mod 1249 = 113 or 1164 mod 1277 = 1276 mod 1277 = 479 or 810 mod 1289 = 1288 mod 1289 = 36 or 1261 mod 1297 = 1296 mod 1297 = 51 or 1250 mod 1301 = 1300 mod 1301 = 257 or 1064 mod 1321 = 1320 mod 1321 = 614 or 747 mod 1361 = 1360 mod 1361 = 668 or 705 mod 1373 = 1372 mod 1373 = 366 or 1015 mod 1381 = 1380 mod 1381 = 452 or 957 mod 1409 = 1408 mod 1409 = 620 or 809 mod 1429 = 1428 mod 1429 = 542 or 891 mod 1433 = 1432 mod 1433 = 497 or 956 mod 1453 = 1452 mod 1453 = 465 or 1016 mod 1481 = 1480 mod 1481 = 225 or 1264 mod 1489 = 1488 mod 1489 = 432 or 1061 mod 1493 = 1492 mod 1493 = 88 or 1461 mod 1549 = 1548 mod 1549 = 339 or 1214 mod 1553 = 1552 mod 1553 = 610 or 987 mod 1597 = 1596 mod 1597 = 40 or 1561 mod 1601 = 1600 mod 1601 = 523 or 1086 mod 1609 = 1608 mod 1609 = 127 or 1486 mod 1613 = 1612 mod 1613 = 166 or 1455 mod 1621 = 1620 mod 1621 = 316 or 1321 mod 1637 = 1636 mod 1637 = 783 or 874 mod 1657 = 1656 mod 1657 = 220 or 1449 mod 1669 = 1668 mod 1669 = 92 or 1601 mod 1693 = 1692 mod 1693 = 414 or 1283 mod 1697 = 1696 mod 1697 = 390 or 1319 mod 1709 = 1708 mod 1709 = 473 or 1248 mod 1721 = 1720 mod 1721 = 410 or 1323 mod 1733 = 1732 mod 1733 = 59 or 1682 mod 1741 = 1740 mod 1741 = 713 or 1040 mod 1753 = 1752 mod 1753 = 775 or 1002 mod 1777 = 1776 mod 1777 = 724 or 1065 mod 1789 = 1788 mod 1789 = 824 or 977 mod 1801 = 1800 mod 1801 = 61 or 1800 mod 1861 = 1860 mod 1861 = 737 or 1136 mod 1873 = 1872 mod 1873 = 137 or 1740 mod 1877 = 1876 mod 1877 = 331 or 1558 mod 1889 = 1888 mod 1889 = 218 or 1683 mod 1901 = 1900 mod 1901 = 712 or 1201 mod 1913 = 1912 mod 1913 = 598 or 1335 mod 1933 = 1932 mod 1933 = 589 or 1360 mod 1949 = 1948 mod 1949 = 259 or 1714 mod 1973 = 1972 mod 1973 = 834 or 1159 mod 1993 = 1992 mod 1993 = 412 or 1585 mod 1997 = 1996 mod 1997 = 229 or 1788 mod 2017 = 2016 mod 2017 = 992 or 1037 mod 2029 = 2028 mod 2029 = 511 or 513 mod 1024 = 1025 mod 2048
There are other k's excluded from the Riesel/Sierpinski problems (Riesel is still much more such k's)

* R30 k=1369:

for even n let n=2*q; factors to: (37*30^q - 1) * (37*30^q + 1)

odd n: covering set 7, 13, 19

* R88 k=400:

for even n let n=2*q; factors to: (20*88^q - 1) * (20*88^q + 1)

odd n: covering set 3, 7, 13

* R95 k=324:

for even n let n=2*q; factors to: (18*95^q - 1) * (18*95^q + 1)

odd n: covering set 7, 13, 229

* S55 k=2500:

odd n: factor of 7

n = = 2 mod 4: factor of 17

n = = 0 mod 4: let n=4q and let m=5*55^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S200 k=16:

odd n: factor of 3

n = = 0 mod 4: factor of 17

n = = 2 mod 4: let n = 4*q - 2 and let m = 20^q*10^(q-1); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S225 k=114244:

for even n let k=4*q^4 and let m=q*15^(n/2); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

odd n: factor of 113

* R10 k=343:

n = = 1 mod 3: factor of 3

n = = 2 mod 3: factor of 37

n = = 0 mod 3: let n=3q and let m=7*10^q; factors to: (m - 1) * (m^2 + m + 1)

* R957 k=64:

n = = 1 mod 3: factor of 73

n = = 2 mod 3: factor of 19

n = = 0 mod 3: let n=3q and let m=4*957^q; factors to: (m - 1) * (m^2 + m + 1)

* S63 k=3511808:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3q and let m=152*63^q; factors to: (m + 1) * (m^2 - m + 1)

* S63 k=27000000:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3q and let m=300*63^q; factors to: (m + 1) * (m^2 - m + 1)

* R936 k=64:

n = = 0 mod 2: let n = 2q; factors to: (8*936^q - 1) * (8*936^q + 1)

n = = 0 mod 3: let n=3q; factors to: (4*936^q - 1) * [16*936^(2q) + 4*936^q + 1]

n = = 1 mod 6: factor of 37

n = = 5 mod 6: factor of 109

Last fiddled with by sweety439 on 2020-07-08 at 18:01   2020-07-08, 18:05   #872
sweety439

Nov 2016

3×5×137 Posts Quote:
 Originally Posted by sweety439 There are other k's excluded from the Riesel/Sierpinski problems (Riesel is still much more such k's) * R30 k=1369: for even n let n=2*q; factors to: (37*30^q - 1) * (37*30^q + 1) odd n: covering set 7, 13, 19 * R88 k=400: for even n let n=2*q; factors to: (20*88^q - 1) * (20*88^q + 1) odd n: covering set 3, 7, 13 * R95 k=324: for even n let n=2*q; factors to: (18*95^q - 1) * (18*95^q + 1) odd n: covering set 7, 13, 229 * S55 k=2500: odd n: factor of 7 n = = 2 mod 4: factor of 17 n = = 0 mod 4: let n=4q and let m=5*55^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1) * S200 k=16: odd n: factor of 3 n = = 0 mod 4: factor of 17 n = = 2 mod 4: let n = 4*q - 2 and let m = 20^q*10^(q-1); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1) * S225 k=114244: for even n let k=4*q^4 and let m=q*15^(n/2); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1) odd n: factor of 113 * R10 k=343: n = = 1 mod 3: factor of 3 n = = 2 mod 3: factor of 37 n = = 0 mod 3: let n=3q and let m=7*10^q; factors to: (m - 1) * (m^2 + m + 1) * R957 k=64: n = = 1 mod 3: factor of 73 n = = 2 mod 3: factor of 19 n = = 0 mod 3: let n=3q and let m=4*957^q; factors to: (m - 1) * (m^2 + m + 1) * S63 k=3511808: n = = 1 mod 3: factor of 37 n = = 2 mod 3: factor of 109 n = = 0 mod 3: let n=3q and let m=152*63^q; factors to: (m + 1) * (m^2 - m + 1) * S63 k=27000000: n = = 1 mod 3: factor of 37 n = = 2 mod 3: factor of 109 n = = 0 mod 3: let n=3q and let m=300*63^q; factors to: (m + 1) * (m^2 - m + 1) * R936 k=64: n = = 0 mod 2: let n = 2q; factors to: (8*936^q - 1) * (8*936^q + 1) n = = 0 mod 3: let n=3q; factors to: (4*936^q - 1) * [16*936^(2q) + 4*936^q + 1] n = = 1 mod 6: factor of 37 n = = 5 mod 6: factor of 109
Of course,

* In the Riesel case if k and b are both r-th powers for an r>1

* In the Sierpinski case if k and b are both r-th powers for an odd r>1

* In the Sierpinski case if k is of the form 4*m^4, and b is 4th power

Then this k proven composite by full algebraic factors   2020-07-08, 18:10 #873 sweety439   Nov 2016 3·5·137 Posts The k's that make a full covering set with all or partial algebraic factors are excluded from the conjectures, unless they also have a covering set of primes (e.g. R4 k=361, R8 k=343, R9 k=49, R16 k=100, R49 k=81, R121 k=100, R125 k=8, S125 k=8, R256 k=100, R1024 k=81), thus cannot be the CK   2020-07-08, 18:16   #874
sweety439

Nov 2016

3·5·137 Posts Quote:
 Originally Posted by sweety439 Of course, * In the Riesel case if k and b are both r-th powers for an r>1 * In the Sierpinski case if k and b are both r-th powers for an odd r>1 * In the Sierpinski case if k is of the form 4*m^4, and b is 4th power Then this k proven composite by full algebraic factors
Also, in the Sierpinski case if b = q^m, k = q^r, where q is not of the form t^s with odd s>1, and m and r have no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution, then this k has no possible prime. (such k's are also excluded from the conjectures)

Examples:

b = q^7, k = q^r, where r = 3, 5, 6 (mod 7).
b = q^14, k = q^r, where r = 6, 10, 12 (mod 14).
b = q^15, k = q^r, where r = 7, 11, 13, 14 (mod 15).
b = q^17, k = q^r, where r = 3, 5, 6, 7, 10, 11, 12, 14 (mod 17).
b = q^21, k = q^r, where r = 5, 10, 13, 17, 19, 20 (mod 21)
b = q^23, k = q^r, where r = 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 (mod 23)
b = q^28, k = q^r, where r = 12, 20, 24 (mod 28)
b = q^30, k = q^r, where r = 14, 22, 26, 28 (mod 30)
b = q^31, k = q^r, where r = 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 (mod 31)
b = q^33, k = q^r, where r = 5, 7, 10, 13, 14, 19, 20, 23, 26, 28 (mod 33)
etc.

(these are all examples for m<=33)

Last fiddled with by sweety439 on 2020-07-10 at 06:18   2020-07-08, 18:22 #875 Uncwilly 6809 > 6502   """"""""""""""""""" Aug 2003 101×103 Posts 43×191 Posts There is no need to quote an entire previous post. Trim your quotes.   2020-07-10, 06:31   #876
sweety439

Nov 2016

3·5·137 Posts Quote:
 Originally Posted by sweety439 Reserve some 1k bases (S17, S27, S51, S56, R38, R54)
See https://en.wikipedia.org/w/index.php...ldid=966555963   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post sweety439 sweety439 20 2020-07-03 17:22 sweety439 sweety439 10 2018-12-14 21:59 sweety439 sweety439 12 2017-12-01 21:56 robert44444uk Conjectures 'R Us 139 2007-12-17 05:17 rogue Conjectures 'R Us 11 2007-12-17 05:08

All times are UTC. The time now is 13:35.

Fri Jul 10 13:35:03 UTC 2020 up 107 days, 11:08, 2 users, load averages: 2.14, 2.07, 1.76