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2020-10-14, 12:53   #34
Dr Sardonicus

Feb 2017
Nowhere

4,517 Posts

Quote:
Originally Posted by Alfred
Quote:
 Originally Posted by Dr Sardonicus It is intuitively obvious that, if k is "sufficiently large", the smallest "2-brilliant" number n > 22k is n = p1*p2, where p1 = nextprime(2k) and p2 = nextprime(p1 + 1). Numerical evidence suggests that "sufficiently large" is k > 3. This notion "obviously" applies to any base.
Dr Sardonicus,

does this statement apply to largest 2-brilliant numbers in base 10?

If yes, please give an example.
No. The argument I had in mind only applies to the smallest 2-brilliant number N > b2k, an even power of the base b > 1. The argument is a triviality. If

N = p*q, p < q prime

with b2k < N < b2k+1

then, in order that p and q have the same number of base-b digits, it is necessary that p >= bk. (The only possibility of equality is with k = 1 when the base b is prime.)

The only question that arises is whether the indicated number p1p2 is less than b2k+1. I believe this will be true for sufficiently large k.

I see no analogous argument for least brilliant greater than an odd power of the base, or for greatest brilliant number less than any power of the base.

 2020-10-14, 13:59 #35 Alfred     May 2013 Germany 83 Posts Dr Sardonicus, thank you for your explanation.
2020-11-01, 15:46   #36
Branger

Oct 2018

23·3 Posts

Continuing this work for 10^169-c, I have found that,

10^169-14319 =
2093963760229909907466815025292144577767961972509185032132596865267781491968551925027 *
4775631837535734107517020048684519409802862518997809812035307071144108182496827007803

I also attach proof files for some of the work I have done, for 10^n+-c for n = 167 and 169. Every number that has a factor larger than 1000 has that factor listed in the files. I'll post the files for n=165 shortly, I seem to have lost some ECM work that I'll redo first.

I intend to continue with n=171, but now I'm starting to get into the territory where the SNFS polynomials are getting rather large coefficients for the batch factorization approach and the relations I already have saved. I'm not sure if it would be quicker to sieve again for a new shared rational side, or if using the bad polynomials with the already existing relations is the least work, but for now I'm using what I have.
Attached Files
 Brilliant_factored_169plus.txt (59.0 KB, 63 views) Brilliant_factored_169minus.txt (32.3 KB, 70 views) Brilliant_factored_167plus.txt (14.4 KB, 60 views) Brilliant_factored_167minus.txt (92.3 KB, 57 views)

 2020-11-01, 18:42 #37 Alfred     May 2013 Germany 83 Posts I think it is a good idea to share these informations. Thank you. So anyone who is interested in can doublecheck the correctness of the statements easily. Last fiddled with by Alfred on 2020-11-01 at 18:45
2020-11-02, 19:50   #38
Branger

Oct 2018

23×3 Posts

And finally here are the proof files for 10^165+-c.
Attached Files
 Brilliant_factored_165minus.txt (64.6 KB, 60 views) Brilliant_factored_165plus.txt (47.7 KB, 71 views)

 2020-11-10, 04:04 #39 swishzzz   Jan 2012 Toronto, Canada 3B16 Posts I am reserving 10^199+c to find the smallest 200-digit number which splits into p100*p100. Likely to take at least a few months with an expected 160+ SNFS factorizations, thought I'd at least post here to prevent any potential duplicated efforts. If anyone is interested in crunching a few of these let me know, I can coordinate sieving efforts on another thread.
2020-11-13, 20:52   #40
Branger

Oct 2018

23·3 Posts

The next one was quicker and only required 27 SNFS factorizations.

10^171+7467 =

15982339170654488061693029140006521400812407348641102533477071444640746972955602480993 *
62569063847432371483112919249240694575724386807642240564815844097979821472243476190219

Continuing with 10^171-c.
Attached Files
 Brilliant_factored_171plus.txt (16.0 KB, 58 views)

2020-12-10, 09:06   #41
Branger

Oct 2018

23·3 Posts

Another 60 SNFS factorizations revealed that

10^171-16569 =
10026073074372053022855343749617316836566548448825765741868691467529514155418582501607 *
99739947293634841017115301301296053264053136150611286806831026018776115664482913987233

Proof file is attached. The batch SNFS approach was still faster than regular SNFS but its getting close, I'll probably continue with 10^173 +- c
Attached Files
 Brilliant_factored_171minus.txt (38.3 KB, 45 views)

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