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2020-09-18, 18:58   #980
sweety439

Nov 2016

7×11×31 Posts

Quote:
 Originally Posted by sweety439 The case for R40 k=490, since all odd n have algebra factors, we only consider even n: n-value : factors 2 : 3^3 · 9679 4 : 43 · 79 · 83 · 1483 6 : 881 · 759379493 8 : 3 · 356807111111111 10 : 31 · 67883 · 813864335521 12 : 53 · 51703370062893081761 18 : 163 · 68860007363271983640081799591 22 : 4801 · 23279 · 3561827 · 4036715519 · 17881240410679 28 : 210323 · 6302441 · 88788971627962097615055082730651231 30 : 38270136643 · 4920560231486977484668641122451121981831 and it does not appear to be any covering set of primes, so there must be a prime at some point. R40 also has two special remain k: 520 and 11560, 520 = 13 * base, 11560 = 289 * base, and the further searching for k = 11560 is 289 with odd n > 1 Another base is R106, which has many k with algebra factors: 64 = 2^6 (thus, all n == 0 mod 2 and all n == 0 mod 3 have algebra factors) 81 = 3^4 (thus, all n == 0 mod 2 have algebra factors) 400 = 20^2 (thus, all n == 0 mod 2 have algebra factors) 676 = 26^2 (thus, all n == 0 mod 2 have algebra factors) 841 = 29^2 (thus, all n == 0 mod 2 have algebra factors) 1024 = 2^10 (thus, all n == 0 mod 2 and all n == 0 mod 5 have algebra factors) We should check whether they have covering set for the n which do not have algebra factors, like the case for R88 k=400 and R30 k=1369
We consider (289*40^n-1)/3 (which is prime for n=1, but there may be covering set for n>1 (and the prime for n=1 (i.e. 3853) must be in the covering set), we should check it: (k=289 for odd n):

n-value : factors
3 : 3^2 · 317 · 2161
5 : 37 · 601 · 443609
7 : 71 · 222299342723
9 : 3 · 8417735111111111
11 : 521 · 77553029814459373
13 : 1093 · 135966569 · 435014942249
17 : 173 · 1201 · 796539523771295275773721
19 : 199 · 827 · 125878441037<12> · 12782225695980733
23 : 31 · 37 · 4493 · 131539610664636811448698039308523
25 : 1693 · 14071 · 83071 · 2786867 · 196665766270295693879723
29 : 43 · 15523495249 · 366735559693 · 11342410093643652930353483
31 : 271 · 1471 · 11144340056387535855201380021957935418919111013
35 : 1289 · (a 55-digit prime)
47 : 207551 · 510199 · 2088787 · (a 60-digit prime)
49 : 15240209 · 10666161587 · 167148848268429277 · (a 47-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

Last fiddled with by sweety439 on 2020-09-18 at 20:39

 2020-09-18, 19:14 #981 sweety439     Nov 2016 7×11×31 Posts For the case for R106: k = 64: since 64 is square and cube, all even n and all n divisible by 3 have algebra factors, and we only want to know whether it has a covering set of primes for all n == 1 or 5 (mod 6), if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture. n-value : factors 1 : 17 · 19 5 : 7 · 17 · 13669 · 25073 7 : 19 · 739 · 32636508923 11 : 105137 · 710341 · 774645021719 13 : 17 · 19 · 2012493124713603631831681 17 : 17 · 16036907 · 301016884615451673389616697 19 : 7 · 19 · 81929 · 1441051 · 1392403219 · 42173384412226351 23 : 4691 · 240422191 · 359534531 · 287087966317907212195482133 35 : 241 · 389 · 39161 · 3351132509456839 · (a 47-digit prime) 47 : 7 · 421 · 17069162801611 · 14667444266312619953 · (a 60-digit prime) 59 : 487 · (a 118-digit composite without known prime factor) 71 : 4289 · 10093 · (a 137-digit composite without known prime factor) Although this number is divisible by 17 for all n == 1 mod 4 and by 19 for all n == 1 mod 6 (which makes this k-value very low weight, since only n == 11 mod 12 can be prime), but it does not appear to be any covering set of primes, so there must be a prime at some point. k = 81: since 81 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture. n-value : factors 1 : 17 · 101 3 : 67 · 287977 5 : 17 · 431 · 727 · 40699 7 : 857 · 2842334911979 11 : 883 · 347963521 · 1000887146689 15 : 47 · 1359940313999 · 607414685128749427 19 : 5 · 1049 · 3331 · 1861172051723 · 150736978974366072719 23 : 6637 · 74623 · 45940781149 · 27196124333848915407481172821 27 : 2135773 · 2196601133149 · 16652026043310698243659019628892454299 31 : 367 · 3894307 · (a 55-digit prime) 35 : 12589419042703 · 73042126655937895819733 · 1354070261224865451982856575186891049 and it does not appear to be any covering set of primes, so there must be a prime at some point. k = 400: since 400 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture. n-value : factors 1 : 3 · 673 3 : 19 · 743 · 1607 5 : 179 · 1424022961 7 : 3 · 4657 · 23917 · 8571317 9 : 19 · 1693713242107962001 11 : 47^2 · 19991 · 8187946182350101 17 : 3362709722608729 · 152528509553573862011 23 : 10889 · 66817096529447428049947387228178558168776171 29 : 67 · 2445989705956469367060937 · 6297691198803985156665528870701561 35 : 34352269373675266693 · 889339893798719344479307 · 47920658139709491455114469269 47 : 607 · (a 94-digit composite without known prime factor) Although this number is divisible by 3 for all n == 1 mod 6 and by 19 for all n == 3 mod 6 (which makes this k-value low weight, since only n == 5 mod 12 can be prime), but it does not appear to be any covering set of primes, so there must be a prime at some point. Last fiddled with by sweety439 on 2020-09-18 at 19:15
 2020-09-18, 20:38 #982 sweety439     Nov 2016 95316 Posts k = 676: since 676 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture. n-value : factors 1 : 17 · 281 3 : 3 · 277 · 64591 5 : 5 · 17 · 19 · 67 · 5573621 7 : 1949 · 3476839593221 9 : 3^3 · 17 · 3271 · 50712496951637 11 : 19 · 42937 · 2432147 · 431166327217 13 : 17 · 1373 · 6351547249 · 64838460350149 17 : 17 · 19 · 61 · 61591784776543827671882518345783 19 : 6299 · 756585273193 · 2861128642099661938794059 23 : 19 · 98443 · 920347627017000007051307391604325416676033 43 : (a 89-digit composite with no known prime factor) 67 : 2843 · (a 134-digit composite with no known prime factor) 79 : 1129 · 32491 · (a 155-digit composite with no known prime factor) 91 : 105899 · (a 181-digit composite with no known prime factor) Although this number is divisible by 3 for all n == 3 mod 6 and by 19 for all n == 5 mod 6 and by 17 for all n == 1 mod 4 (which makes this k-value very low weight, since only n == 7 mod 12 can be prime), but it does not appear to be any covering set of primes, so there must be a prime at some point. k = 841: since 841 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture. n-value : factors 1 : 3 · 283 3 : 271 · 35201 5 : 61 · 677 · 2595479 7 : 3^2 · 5^2 · 5352605493383 9 : 4679 · 8663 · 333839809991 11 : 67 · 11440889 · 198352025576693 15 : 359487408541 · 53396278847280064403 17 : 5 · 19927 · 140909 · 15362282538731494849528849 21 : 271 · 7457 · 663563 · 20305527277370848392217057350779 23 : 94547 · 534824108672537 · 6050383020924045192372407269 29 : 84737 · (a 55-digit prime) 33 : 311 · 1888306597 · 1129552782935923 · 2923571188269551 · 28251866661502752658291361 51 : 2843 · (a 101-digit composite with no known prime factor) 53 : 13456811 · 88286677 · 6437291630956799 · (a 78-digit prime) 59 : (a 121-digit composite with no known prime factor) 63 : 2371 · 6059059478263861 · (a 110-digit composite with no known prime factor) and it does not appear to be any covering set of primes, so there must be a prime at some point. k = 1024: since 1024 is square and 5-th power, all even n and all n divisible by 5 have algebra factors, and we only want to know whether it has a covering set of primes for all n == 1, 3, 7, 9 (mod 10), if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture. n-value : factors 1 : 97 · 373 3 : 17 · 3407 · 7019 7 : 17 · 3019053696484613 9 : 647 · 3581827 · 248841380929 11 : 3 · 17^2 · 7473501436891484179943 13 : 449 · 1447 · 112057280449127255045987 17 : 3^2 · 406591 · 2126171 · 1181353712721405831409129 19 : 17 · 283 · 258373 · 9179867 · 9050472811960369021895401 21 : 1831 · 972605267 · 1597539586927967 · 407873305308400559 33 : 1223 · (a 67-digit prime) 37 : 7753 · 2460302303 · (a 65-digit prime) 49 : 97 · 839 · 25561 · 136811 · 45385621130173559982180883 · (a 62-digit prime) 57 : 1777 · 66191 · 10482163 · 4863222893 · (a 94-digit prime) 61 : (a 127-digit composite with no known prime factor) and it does not appear to be any covering set of primes, so there must be a prime at some point. Last fiddled with by sweety439 on 2020-09-20 at 21:58
 2020-09-20, 22:11 #983 sweety439     Nov 2016 7·11·31 Posts For R40, these remain k's have algebra factors: * k=490 (=square*base/4), all odd n are algebraic * k=11560 (=289*base, 289 itself was eliminated at n=1), all odd n are algebraic * k=12250 (=square*base/4), all odd n are algebraic * k=12544 (=square), all even n are algebraic * k=15376 (=square), all even n are algebraic For R52, these remain k's have algebra factors: * k=21316 (=square), all even n are algebraic For R78, these remain k's have algebra factors: * k=4489 (=square), all even n are algebraic * k=7800 (=100*base, 100 itself was eliminated at n=1), all odd n are algebraic * k=8649 (=square), all even n are algebraic * k=12167 (=cube), all n divisible by 3 are algebraic * k=13824 (=cube), all n divisible by 3 are algebraic * k=59536 (=square), all even n are algebraic For R96, these remain k's have algebra factors: * k=1681 (=square), all even n are algebraic * k=5046 (=square*base/16), all odd n are algebraic * k=9216 (=1*base^2, 1 itself was eliminated at n=2), all n such that n+2 is composite are algebraic * k=16641 (=square), all even n are algebraic For R106, these remain k's have algebra factors: * k=64 (=square and cube), all even n and all n divisible by 3 are algebraic * k=81 (=square), all even n are algebraic * k=400 (=square), all even n are algebraic * k=676 (=square), all even n are algebraic * k=841 (=square), all even n are algebraic * k=1024 (=square and 5th power), all even n and all n divisible by 5 are algebraic * k=2116 (=square), all even n are algebraic * k=3136 (=square), all even n are algebraic * k=3481 (=square), all even n are algebraic * k=4096 (=square and cube), all even n and all n divisible by 3 are algebraic * k=5776 (=square), all even n are algebraic * k=7744 (=square), all even n are algebraic * k=10816 (=square), all even n are algebraic * k=12321 (=square), all even n are algebraic For R124, these remain k's have algebra factors: * k=441 (=square), all even n are algebraic * k=1156 (=square), all even n are algebraic * k=1519 (=square*base/4), all odd n are algebraic * k=4096 (=square and cube), all even n and all n divisible by 3 are algebraic * k=7396 (=square), all even n are algebraic Last fiddled with by sweety439 on 2020-09-23 at 19:58
 2020-09-20, 22:12 #984 sweety439     Nov 2016 1001010100112 Posts For Riesel base b, a k-value has algebra factors if and only if there exists n such that k*b^n is perfect power For Sierpinski base b, a k-value has algebra factors if and only if there exists n such that k*b^n is either perfect odd power or of the form 4*m^4 Last fiddled with by sweety439 on 2020-09-20 at 22:13
 2020-09-22, 20:52 #985 sweety439     Nov 2016 7·11·31 Posts Riesel case: (k*b^n-1)/gcd(k-1,b-1) Sierpinski case: (k*b^n+1)/gcd(k+1,b-1) If k is not rational power of b, then: * In Riesel case, (k*b^n-1)/gcd(k-1,b-1) has algebra factors if and only if k*b^n is perfect power (of the form m^r with r>1) * In Sierpinski case, (k*b^n+1)/gcd(k+1,b-1) has algebra factors if and only if k*b^n is either perfect odd power (of the form m^r with odd r>1) or of the form 4*m^4 If k is rational power of b (let k = m^r, b = m^s): * In Riesel case, (k*b^n-1)/gcd(k-1,b-1) has algebra factors if and only if n*s+r is composite * In Sierpinski case, (k*b^n+1)/gcd(k+1,b-1) has algebra factors if and only if n*s+r is (not power of 2, if valuation(r,2) >= valuation(s,2)) (not of the form p*2^valuation(r,2) with p prime, if valuation(r,2) < valuation(s,2)) Last fiddled with by sweety439 on 2020-09-23 at 19:56
2020-09-22, 22:42   #986
sweety439

Nov 2016

238710 Posts

Quote:
 Originally Posted by sweety439 For R40, these remain k's have algebra factors: * k=490 (=square*base/4), all odd n are algebraic * k=11560 (=289*base, 289 itself was eliminated at n=1), all odd n are algebraic * k=12250 (=square*base/4), all odd n are algebraic * k=12544 (=square), all even n are algebraic * k=15376 (=square), all even n are algebraic For R52, these remain k's have algebra factors: * k=21316 (=square), all even n are algebraic For R78, these remain k's have algebra factors: * k=4489 (=square), all even n are algebraic * k=7800 (=100*base, 100 itself was eliminated at n=1), all odd n are algebraic * k=8649 (=square), all even n are algebraic * k=12167 (=cube), all n divisible by 3 are algebraic * k=13824 (=cube), all n divisible by 3 are algebraic * k=59536 (=square), all even n are algebraic For R96, these remain k's have algebra factors: * k=1681 (=square), all even n are algebraic * k=5046 (=square*base/16), all odd n are algebraic * k=9216 (=1*base^2, 1 itself was eliminated at n=2), all n such that n+2 is not prime are algebraic * k=16641 (=square), all even n are algebraic For R106, these remain k's have algebra factors: * k=64 (=square and cube), all even n and all n divisible by 3 are algebraic * k=81 (=square), all even n are algebraic * k=400 (=square), all even n are algebraic * k=676 (=square), all even n are algebraic * k=841 (=square), all even n are algebraic * k=1024 (=square and 5th power), all even n and all n divisible by 5 are algebraic * k=2116 (=square), all even n are algebraic * k=3136 (=square), all even n are algebraic * k=3481 (=square), all even n are algebraic * k=4096 (=square and cube), all even n and all n divisible by 3 are algebraic * k=5776 (=square), all even n are algebraic * k=7744 (=square), all even n are algebraic * k=10816 (=square), all even n are algebraic * k=12321 (=square), all even n are algebraic For R124, these remain k's have algebra factors: * k=441 (=square), all even n are algebraic * k=1156 (=square), all even n are algebraic * k=1519 (=square*base/4), all odd n are algebraic * k=4096 (=square and cube), all even n and all n divisible by 3 are algebraic * k=7396 (=square), all even n are algebraic
R88 have more such examples, see post https://mersenneforum.org/showpost.p...&postcount=486

 2020-09-22, 22:56 #987 sweety439     Nov 2016 7·11·31 Posts Conjecture: For integer triple (k,b,c), k>=1, b>=2, c != 0, gcd(k,c)=1, gcd(b,c)=1, there is a prime of the form (k*b^n+c)/gcd(k+c,b-1) with integer n>=1 if and only if (k*b^n+c)/gcd(k+c,b-1) has no covering set (including: covering set of fixed prime factors or covering set of all algebra factors or full covering set of partial algebra factors and partial fixed prime factors). Last fiddled with by sweety439 on 2020-09-23 at 01:45
 2020-09-23, 20:02 #988 sweety439     Nov 2016 7×11×31 Posts See https://github.com/xayahrainie4793/S...-variable-base for the status of 1<=k<=12 and 2<=b<=1024, all searched to n>=6000 (n>=100000 for gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) = 1, n>=100000 for Riesel k=1, n>=8388607 for Sierpinski k=1 and b even, n>=524287 for Sierpinski k=1 and b odd) Remain bases: Riesel k=1: {185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015} (totally 30 bases) Riesel k=2: {581, 992, 1019} (totally 3 bases) Riesel k=3: {347, 575, 587, 588, 595, 659, 699, 711, 731, 751, 763, 783, 795, 823, 972} (totally 15 bases) Riesel k=4: {178, 223, 271, 275, 310, 373, 412, 438, 475, 535, 647, 650, 653, 655, 718, 727, 742, 751, 778, 790, 812, 862, 868, 871, 898, 927, 940, 968, 970, 997, 1003} (totally 31 bases) Last fiddled with by sweety439 on 2020-09-24 at 00:47
 2020-09-23, 21:16 #989 sweety439     Nov 2016 45238 Posts Riesel k=5: {31, 117, 181, 338, 411, 429, 489, 499, 535, 581, 583, 631, 717, 757, 998} (totally 15 bases) Riesel k=6: {234, 412, 549, 553, 573, 619, 750, 878, 894, 954, 986} (totally 11 bases) Riesel k=7: {202, 233, 308, 373, 392, 398, 437, 463, 518, 548, 638, 662, 713, 807, 821, 848, 878, 893, 895, 953, 1015} (totally 21 bases) Riesel k=8: {321, 328, 372, 374, 407, 432, 477, 575, 665, 680, 697, 710, 721, 722, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972, 974} (totally 25 bases) Last fiddled with by sweety439 on 2020-09-24 at 00:49
 2020-09-23, 23:13 #990 sweety439     Nov 2016 7×11×31 Posts Riesel k=9: {107, 207, 237, 325, 347, 378, 438, 483, 536, 566, 570, 592, 636, 688, 705, 711, 718, 823, 830, 835, 852, 893, 907, 926, 927, 995, 1010} (totally 27 bases) Riesel k=10: {80, 233, 262, 284, 307, 505, 530, 551, 611, 691, 712, 724, 883, 899, 912, 980} (totally 16 bases) Riesel k=11: {65, 123, 137, 163, 173, 207, 214, 221, 227, 235, 247, 263, 283, 293, 317, 331, 375, 377, 422, 444, 452, 458, 471, 487, 533, 542, 555, 603, 627, 638, 663, 668, 691, 723, 752, 793, 804, 823, 843, 857, 863, 872, 907, 911, 923, 949, 950, 962, 987} (totally 49 bases) Riesel k=12: {263, 615, 912, 978} (totally 4 bases) Last fiddled with by sweety439 on 2020-09-24 at 00:50

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