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2018-12-31, 18:11   #67
10metreh

Nov 2008

2×33×43 Posts

Quote:
 Originally Posted by MisterBitcoin Well, at first i guessed I found the largest ck, but. It´s not that easy because n=1 and n=14 are prime; so the guessed number is not the ck.
covering.exe has a bug when searching very large k. There is a version bigcovering.exe which fixes this bug - it's available from the same site as covering.exe.

The likely CK for this base is 1429458644604553 which has covering set {19, 37, 61, 307, 1051, 39916801} with period 18. The reason it's so big is that the primes that can appear in the covering set with period d are the primes p such that b has order d mod p; these are prime factors of the cyclotomic number Φd(b). If for small d these numbers have lots of small prime factors then there are lots of primes that are likely to appear as factors and so the CK will be small. If there are few small factors then we will either need a big prime in the covering set (making the CK big as well) or a lot of small primes with longer periods (which will probably also make the CK big).

For b = 11!, there are very few small factors:
Φ2(b) = b+1 = 39916801
Φ3(b) = b^2+b+1 = 61*26120507576341
Φ4(b) = b^2+1 = 1593350922240001
Φ5(b) = b^4+b^3+b^2+b+1 = 761*3336093593961274918315629641
Φ6(b) = b^2-b+1 = 1051*1516033189651
So it's not a surprise that the CK is huge.

Quote:
 I´ve only tryed exponent 144 (above is seen using 72; for sure. :P ). Is there any other exponents that I might try?
If Φ5(b) or Φ10(b) have lots of small factors then it's worth trying periods with 5 as a factor, but that isn't the case here.

Last fiddled with by 10metreh on 2018-12-31 at 18:18 Reason: small correction

 2019-01-01, 18:24 #68 lalera     Jul 2003 11178 Posts hi, i am starting R2019 CK 304
2019-01-07, 10:14   #69
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

10110010110102 Posts

Quote:
 Originally Posted by 10metreh covering.exe has a bug when searching very large k. There is a version bigcovering.exe which fixes this bug - it's available from the same site as covering.exe. The likely CK for this base is 1429458644604553 which has covering set {19, 37, 61, 307, 1051, 39916801} with period 18. The reason it's so big is that the primes that can appear in the covering set with period d are the primes p such that b has order d mod p; these are prime factors of the cyclotomic number Φd(b). If for small d these numbers have lots of small prime factors then there are lots of primes that are likely to appear as factors and so the CK will be small. If there are few small factors then we will either need a big prime in the covering set (making the CK big as well) or a lot of small primes with longer periods (which will probably also make the CK big). For b = 11!, there are very few small factors: Φ2(b) = b+1 = 39916801 Φ3(b) = b^2+b+1 = 61*26120507576341 Φ4(b) = b^2+1 = 1593350922240001 Φ5(b) = b^4+b^3+b^2+b+1 = 761*3336093593961274918315629641 Φ6(b) = b^2-b+1 = 1051*1516033189651 So it's not a surprise that the CK is huge. If Φ5(b) or Φ10(b) have lots of small factors then it's worth trying periods with 5 as a factor, but that isn't the case here.
3036132198346310 is a small improvement.

2019-01-07, 10:59   #70
10metreh

Nov 2008

91216 Posts

Quote:
 Originally Posted by henryzz 3036132198346310 is a small improvement.
It would be if it was actually smaller than 1429458644604553 and had a covering set (I haven't found a prime yet, but what is the factor for n=15?)

2019-01-07, 18:47   #71
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

572210 Posts

Quote:
 Originally Posted by 10metreh It would be if it was actually smaller than 1429458644604553 and had a covering set (I haven't found a prime yet, but what is the factor for n=15?)
Typically I looked at Riesel by mistake. I also miscounted digits.

2019-01-08, 17:32   #72
pepi37

Dec 2011
After milion nines:)

17×79 Posts

Quote:
 Originally Posted by lalera hi, i am starting R2019 CK 304
What is ck of S2019?

2019-01-08, 22:01   #73
pepi37

Dec 2011
After milion nines:)

17·79 Posts

Quote:
 Originally Posted by pepi37 What is ck of S2019?
Ok found it
So I start S2019 :)

2019-01-09, 11:30   #74
lalera

Jul 2003

24F16 Posts

Quote:
 Originally Posted by pepi37 Ok found it So I start S2019 :)
hi,
S2019, ck is 304

2019-01-09, 14:21   #75
pepi37

Dec 2011
After milion nines:)

17×79 Posts

Quote:
 Originally Posted by lalera hi, S2019, ck is 304
Thanks!

 2019-02-01, 15:05 #76 lalera     Jul 2003 10010011112 Posts hi, status update for R2019 range n=1 to 100k done 10 k´s remain for more info visit lalera.alotspace.com continuing
2019-02-02, 09:47   #77
sweety439

Nov 2016

13×173 Posts

Quote:
 Originally Posted by lalera hi, status update for R2019 range n=1 to 100k done 10 k´s remain for more info visit lalera.alotspace.com continuing
For k=4, k=64, and k=144:

Odd n has factor of 5
Even n has algebra factors

For k=100:

Odd n has factor of 101
Even n has algebra factors

Thus R2019 has only 6 k’s remain at n=100K:

84, 114, 204, 242, 296, 302

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