 mersenneforum.org A Sierpinski/Riesel-like problem
 Register FAQ Search Today's Posts Mark Forums Read  2020-09-03, 08:33 #969 sweety439   Nov 2016 23×103 Posts 2 (probable) primes found for R70: (376*70^6484-1)/3 (496*70^4934-1)/3 k=811 still remains ....   2020-09-03, 21:37 #970 sweety439   Nov 2016 23×103 Posts https://docs.google.com/document/d/e...Ns5LlL_FA0/pub Update newest file for Sierpinski problems to include the top 10 (probable) primes for S78, S96, and S126   2020-09-04, 22:43 #971 sweety439   Nov 2016 23·103 Posts https://docs.google.com/document/d/e...MpCV9458Wi/pub Update newest file for Sierpinski conjectures, according to GFN2 and GFN10, the test limit of S512 k=2 is (2^54-1)/9-1 = 2001599834386886, and the test limit of S10 k=100 is 2^31-3 = 2147483645   2020-09-04, 23:03   #972
sweety439

Nov 2016

23×103 Posts Quote:
 Originally Posted by sweety439 S117: k=11 k=47 k=67 k=75 k=77 k=81 (proven by N+1-method) S256: (k=11 is only probable prime) k=23 S1024: k=14 k=41 k=44
R4:

k=106

R7: (k=197 and 367 are only probable primes)

k=79
k=139 (proven by N-1-method) (certificate for large prime factor for N-1)
k=159
k=299
k=313
k=391
k=419
k=429
k=437
k=451

R10:

k=121

R12:

k=298

R17:

k=13
k=29

R26:

k=121

R31:

k=21
k=39 (proven by N-1-method) (certificate for large prime factor for N-1)
k=49
k=113
k=115
k=123
k=124

R33:

k=213

R35:

k=1 (proven by N-1-method)

R37:

k=5 (proven by N-1-method)

R39:

k=1 (proven by N-1-method)

R43:

k=4 (proven by N-1-method)

R45:

k=53

Last fiddled with by sweety439 on 2020-09-05 at 12:03   2020-09-05, 20:17   #973
sweety439

Nov 2016

23×103 Posts Quote:
 Originally Posted by sweety439 Update files.
Update files
Attached Files Sierp k1.txt (7.6 KB, 14 views) Sierp k10.txt (7.6 KB, 15 views) Sierp k12.txt (7.4 KB, 18 views) Riesel k9.txt (12.1 KB, 18 views) Riesel k10.txt (7.5 KB, 15 views)

Last fiddled with by sweety439 on 2020-09-05 at 20:18   2020-09-06, 19:49 #974 sweety439   Nov 2016 23·103 Posts Also reserved R88 and found these (probable) primes: (49*88^2223-1)/3 (79*88^7665-1)/3 (235*88^1330-1)/3 (346*88^2969-1)/3 (541*88^1187-1)/3 (544*88^8904-1)/3 k = 46, 94, 277, 508 are still remaining.   2020-09-06, 20:13 #975 sweety439   Nov 2016 1001010000012 Posts Still no (probable) prime found for R70 k=811 If a (probable) prime for R70 k=811 were found, then will produce a group of 11 consecutive proven Riesel conjectures: R67 to R77   2020-09-06, 23:53   #976
sweety439

Nov 2016

23×103 Posts Quote:
 Originally Posted by sweety439 R4: k=106 R7: (k=197 and 367 are only probable primes) k=79 k=139 (proven by N-1-method) (certificate for large prime factor for N-1) k=159 k=299 k=313 k=391 k=419 k=429 k=437 k=451 R10: k=121 R12: k=298 R17: k=13 k=29 R26: k=121 R31: k=21 k=39 (proven by N-1-method) (certificate for large prime factor for N-1) k=49 k=113 k=115 k=123 k=124 R33: k=213 R35: k=1 (proven by N-1-method) R37: k=5 (proven by N-1-method) R39: k=1 (proven by N-1-method) R43: k=4 (proven by N-1-method) R45: k=53
R46: (k=86, 576, and 561 are only probable primes)

k=100
k=121
k=142
k=256
k=386

R49:

k=79

R51:

k=1

R57:

k=87 (proven by N-1-method) (certificate for large prime factor for N-1)

R58: (k=382, 400, and 421 are only probable primes)

k=4 (proven by N-1-method)
k=103
k=109
k=142
k=163
k=217
k=271
k=334
k=361
k=379
k=445
k=457
k=487

R61: (k=13 is only probable prime)

k=10
k=41
k=77

Last fiddled with by sweety439 on 2020-09-06 at 23:56   2020-09-07, 00:27 #977 sweety439   Nov 2016 23·103 Posts Update newest file for Riesel problems to include recent status for R70 and R88   2020-09-13, 12:37 #978 sweety439   Nov 2016 23·103 Posts I conjectured that: If (k,b,c) is integer triple, k>=1, b>=2, c != 0, gcd(k,c)=1, gcd(b,c)=1, if (k*b^n+c)/gcd(k+c,b-1) does not have covering set of primes for the n such that: (the set of the n satisfying these conditions must be nonempty, or (k*b^n+c)/gcd(k+c,b-1) proven composite by full algebra factors) * if c != +-1, (let r be the largest integer such that (-c) is perfect r-th power) k*b^n is not perfect r-th power * if c = 1, k*b^n is not perfect odd power (of the form m^r with odd r>1), except the case: 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 * if c = -1, k*b^n is not perfect power (of the form m^r with r>1), except the case: 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 * 4*k*b^n*c is not perfect 4-th power Then there are infinitely many primes of the form (k*b^n+c)/gcd(k+c,b-1), except the case: 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. Last fiddled with by sweety439 on 2020-09-13 at 12:39   2020-09-17, 19:28   #979
sweety439

Nov 2016

23·103 Posts Quote:
 Originally Posted by sweety439 Also these cases: S15 k=343: since 343 is cube, 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 not divisible by 3, 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 : 31 · 83 2 : 2^2 · 11 · 877 4 : 2^2 · 809 · 2683 5 : 811 · 160583 7 : 11^2 · 242168453 11 : 31 · 101 · 25357 · 18684739 13 : 397 · 1281101 · 656261029 17 : 11 · 27479311 · 55900668804553 29 : 53 · 197741 · 209188613429183386499227445981 35 : 1337724923 · 18667724069720862256321575167267431 43 : 20943991 · 3055827403675875709696160949928034201885723243 61 : 23539 · (a 61-digit prime) and it does not appear to be any covering set of primes, so there must be a prime at some point. S61 k=324: since 324 is of the form 4*m^4, all n divisible by 4 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 4, 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 : 59 · 67 2 : 41 · 5881 3 : 13 · 1131413 5 : 5 · 7 · 1563709723 6 : 13 · 256809250661 7 : 23 · 1255679 · 7051433 13 : 191 · 7860337 · 27268229 · 256289843 14 : 1540873 · 1698953 · 244480646906833 31 : 1888149043321 · 441337391577139 · 1721840403480692512106884569347 34 : 10601 · 174221 · (a 54-digit prime) and it does not appear to be any covering set of primes, so there must be a prime at some point.
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 k = 289 with odd n > 1 (since 289 is square, all even n for k = 289 have algebra factors)

Another base is R106, which has many k with algebra factors (these k are all squares):

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 R30 k=1369 and R88 k=400

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