Smallest 10^179+c Brilliant Number (p90 * p90) - Page 4

Dr Sardonicus · 2020-10-14, 12:53

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 > 2^2k is n = p₁*p₂, where p₁ = nextprime(2^k) 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 > b^2k, an even power of the base b > 1. The argument is a triviality. If

N = p*q, p < q prime

with b^2k < N < b^2k+1

then, in order that p and q have the same number of base-b digits, it is necessary that p >= b^k. (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 b^2k+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.

Alfred · 2020-10-14, 13:59

Dr Sardonicus,

thank you for your explanation.

Branger · 2020-11-01, 15:46

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.

Alfred · 2020-11-01, 18:42

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.

Branger · 2020-11-02, 19:50

And finally here are the proof files for 10^165+-c.

swishzzz · 2020-11-10, 04:04

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.

Branger · 2020-11-13, 20:52

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

10^171+7467 =

15982339170654488061693029140006521400812407348641102533477071444640746972955602480993 *
62569063847432371483112919249240694575724386807642240564815844097979821472243476190219

Continuing with 10^171-c.

Branger · 2020-12-10, 09:06

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

Branger · 2021-07-19, 08:12

Another 171 SNFS factorizations show that

10^173 + 46323 =
219678541518943592018357235810696621038534586888886266590632787250688693297392179181097 *
455210596850110078552386152139990527441828648981499480797698798803694511821552768572859

Proof file is attached.

I started this trying the factorization factory approach with the relations I had, but there were so many strange faliures that I gave up and ran all as regular SNFS. The factory approach would probably have saved ~30% time per factorization if it had worked, even with the now rather horrible polynomials.

I think I will give 10^221+c a try next, to continue with the factory approach. May as well try for a record while I'm at it

WraithX · 2021-08-06, 19:33

swishzzz asked if I had any data from my previous brilliant number searches. I've compiled all the data I have and attached them in the included zip file. Each subfolder has a number of files each containing the ggnfs factoring results for the "difficult" factorizations, and several of the subfolders have a small_factors.txt file which lists all easy to factor numbers. swishzzz, hopefully this will be useful for your brilliant database.

sweety439 · 2021-08-07, 05:59

You can update OEIS sequences A084475, A084476, A083289, A083128, A083182, also create sequences "least k such that 10^(2*n+1)-k is brilliant number" (1, 11, 27, 189, 137, 357, ...), "least k such that 10^n-k is brilliant number" (1, 51, 11, 591, 27, 5991, 189, 539271), "smallest n-digit brilliant number" (4, 10, 121, 1003, 10201, 100013, 1018081, 10000043, ...), "largest n-digit brilliant number" (9, 49, 989, 9409, 99973, 994009, 9999811, 99460729, ...), "smallest n-digit 4-brilliant number" (16, 100, 1029, 14641, 100529, 1000109, 10005647, ...), "largest n-digit 4-brilliant number" (90, 875, 2401, 99671, 999973, 9991291, 88529281, ...)

2020-11-01, 18:42	#37
Alfred May 2013 Germany 2×43 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

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2020-10-14, 13:59	#35
Alfred May 2013 Germany 2×43 Posts	Dr Sardonicus, thank you for your explanation.

2020-11-10, 04:04	#39
swishzzz Jan 2012 Toronto, Canada 3×31 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.

2021-08-07, 05:59	#44
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2³·7·53 Posts	You can update OEIS sequences A084475, A084476, A083289, A083128, A083182, also create sequences "least k such that 10^(2*n+1)-k is brilliant number" (1, 11, 27, 189, 137, 357, ...), "least k such that 10^n-k is brilliant number" (1, 51, 11, 591, 27, 5991, 189, 539271), "smallest n-digit brilliant number" (4, 10, 121, 1003, 10201, 100013, 1018081, 10000043, ...), "largest n-digit brilliant number" (9, 49, 989, 9409, 99973, 994009, 9999811, 99460729, ...), "smallest n-digit 4-brilliant number" (16, 100, 1029, 14641, 100529, 1000109, 10005647, ...), "largest n-digit 4-brilliant number" (90, 875, 2401, 99671, 999973, 9991291, 88529281, ...)