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Old 2020-09-18, 18:58   #980
sweety439
 
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Originally Posted by sweety439 View Post
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
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