mersenneforum.org Smallest 10^179+c Brilliant Number (p90 * p90)
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 2019-12-15, 17:22 #23 Branger   Oct 2018 2·32 Posts For the next one I had a bit more luck and found it after only 20 SNFS factorizations. 10^167+6453 = 203214913448641292965085614133875784826110271627178496334164562386280018360230767193 * 492089868321958070178727516157409397743940386446977363088474243145123507652778037821
 2020-05-01, 09:55 #24 Branger   Oct 2018 2×32 Posts The next one took much longer, requiring 140 SNFS factorizations, but now I am happy to report that 10^167-38903 = 203295679518280624355545616168150860499969671339902409710914658195811040122874591267 * 491894369014408217255986821288848144293491232238922901468113805403238598987818347491
 2020-08-30, 20:47 #25 Branger   Oct 2018 100102 Posts And an additional 90 SNFS factorizations show that 10^169 + 25831 = 1578640553322706420836164892965282526510795833878698113106432074544532020287270837641 * 6334564241968890714235608069337466422072649801566867783807088391771261962286575780591
 2020-09-13, 14:26 #26 swishzzz   Jan 2012 Toronto, Canada 22·11 Posts Are these recently found brilliant numbers tracked anywhere? https://www.alpertron.com.ar/BRILLIANT.HTM doesn't seem to have anything above 155 digits. Reserving 10^147 - n for n < 10000.
 2020-09-13, 17:02 #27 alpertron     Aug 2002 Buenos Aires, Argentina 2×3×223 Posts At this moment I'm making changes to my calculator that factors and finds the roots of polynomials (you can see it at https://www.alpertron.com.ar/POLFACT.HTM). After that, I will update the page of brilliant numbers. You can select whether you want to appear with your real name or with the username at this forum. Thanks a lot for your efforts.
 2020-09-19, 00:24 #28 alpertron     Aug 2002 Buenos Aires, Argentina 24728 Posts I've just added the discoveries posted to this thread to https://www.alpertron.com.ar/BRILLIANT.HTM and also fixed the errors detected at https://www.alpertron.com.ar/BRILLIANT2.HTM
 2020-09-19, 21:21 #29 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 23×797 Posts Thank you! May I also point you at https://mersenneforum.org/showthread.php?t=22626 ?
 2020-09-25, 03:28 #30 alpertron     Aug 2002 Buenos Aires, Argentina 53A16 Posts I've just updated the page https://www.alpertron.com.ar/BRILLIANT3.HTM with your results. Thanks a lot.
 2020-09-30, 16:03 #31 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 23×797 Posts The smallest 400-bit number with two 200-bit prime factors is 0x98B1A3CA31877A7140FEFFA30608FBAB17232646BEC3BAA167 * 0xD699697AC5B27CD0A75D35F9E19320D82A4F4101B550C65E97 = 2^399+198081 (about 15 curves at b1=1e6 for 2^399+{1..10^6} and then SNFS on about 800 400-bit numbers taking a median of 15740 seconds on one thread of i9/7940X) I've got an evidence file with a prime factor of less than 200 bits for every 2^399+N which is composite and coprime to (2^23)! but am not quite sure where's best to put it Last fiddled with by fivemack on 2020-09-30 at 16:03
2020-10-14, 08:54   #32
Alfred

May 2013
Germany

3·52 Posts

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.

2020-10-14, 12:52   #33
swishzzz

Jan 2012

4410 Posts

2^293 - 33769 is the product of two 147-bit primes:

Quote:
 P45 = 126510626365064224002933822140885711272659161 P45 = 125794520368511128755464888278721782242924143
Factor file for 2^293 - c with c > 0 attached.
Attached Files
 293b_brilliant.txt (33.8 KB, 13 views)

Last fiddled with by swishzzz on 2020-10-14 at 12:55

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