Primes made mostly of nines
10^10000  10^8668  1 is a pseudoprime; can I assert that it's prime because we've got a very boring factorisation of 86.68% of n+1 ?

[QUOTE=fivemack;560683]10^10000  10^8668  1 is a pseudoprime; can I assert that it's prime because we've got a very boring factorisation of 86.68% of n+1 ?[/QUOTE]Why do you care?
It isn't that 10kdigit numbers are difficult to prove prime by ECPP or APRCL these days. 
FactorDB instantly proved it by N+1 as being prime.

[QUOTE=fivemack;560683]10^10000  10^8668  1 is a pseudoprime; can I assert that it's prime because we've got a very boring factorisation of 86.68% of n+1 ?[/QUOTE]
Combined Theorem 1 is enough from [url]https://primes.utm.edu/prove/prove3_3.html[/url] with F1=1, F2=10^8668. 
[QUOTE=xilman;560687]Why do you care?
It isn't that 10kdigit numbers are difficult to prove prime by ECPP or APRCL these days.[/QUOTE] Using a gigahertzmonth of compute for something which can sensibly be asserted by inspection would get probably ruder remarks from you and RDS :) (my housemate had found a tweet getting excited about a 6400digit prime comprised entirely of nines with a single eight, and I thought this was not a particularly exciting result) 
[QUOTE=fivemack;560683]10^10000  10^8668  1 is a pseudoprime; can I assert that it's prime because we've got a very boring factorisation of 86.68% of n+1 ?[/QUOTE]
[CODE]./pfgw64 tp q"10^10000  10^8668  1" T4 PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8] Primality testing 10^10000  10^8668  1 [N+1, BrillhartLehmerSelfridge] Running N+1 test using discriminant 43, base 1+sqrt(43) 10^10000  10^8668  1 is prime! (10.6651s+0.0255s) [/CODE] :smile: Back in the day, we found [URL="https://primes.utm.edu/primes/page.php?id=168"]this one[/URL] when PRP tests took 100 mins each on Athlons at 1GHz. What programs have you been using to find your prime? The following was done on one core of a Haswell at 3.7GHz. [code]cat NRD_gigantic ABC2 10^1000010^$a1 a: from 1 to 9999 [/code] [CODE]time ./pfgw64 N f NRD_gigantic PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8] Recognized ABC Sieve file: ABC2 File ***WARNING! file NRD_gigantic may have already been fully processed. 10^1000010^7501 has factors: 2313617 10^1000010^15891 has factors: 2635553 10^1000010^34861 is 3PRP! (1.1229s+0.0885s) 10^1000010^39091 is 3PRP! (1.0102s+0.1867s) 10^1000010^41511 has factors: 376769 10^1000010^51331 is 3PRP! (1.0614s+0.0897s) 10^1000010^53341 has factors: 772147 10^1000010^61341 has factors: 2749921 10^1000010^79281 is 3PRP! (1.1574s+0.1369s) 10^1000010^80721 has factors: 2742227 10^1000010^86681 is 3PRP! (0.9757s+0.0931s) 10^1000010^87401 has factors: 2600837 real 34m58.010s user 34m57.090s sys 0m0.524s [/CODE] [CODE]./pfgw64 tp q"10^1000010^34861" PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8] Primality testing 10^1000010^34861 [N+1, BrillhartLehmerSelfridge] Running N+1 test using discriminant 7, base 1+sqrt(7) 10^1000010^34861 is prime! (3.8788s+0.0002s) ./pfgw64 tp q"10^1000010^39091" PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8] Primality testing 10^1000010^39091 [N+1, BrillhartLehmerSelfridge] Running N+1 test using discriminant 7, base 1+sqrt(7) 10^1000010^39091 is prime! (3.8218s+0.0001s) ./pfgw64 tp q"10^1000010^51331" PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8] Primality testing 10^1000010^51331 [N+1, BrillhartLehmerSelfridge] Running N+1 test using discriminant 7, base 1+sqrt(7) 10^1000010^51331 is prime! (3.9534s+0.0001s) ./pfgw64 tp q"10^1000010^79281" PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8] Primality testing 10^1000010^79281 [N+1, BrillhartLehmerSelfridge] Running N+1 test using discriminant 7, base 1+sqrt(7) 10^1000010^79281 is prime! (4.5040s+0.0002s) [/CODE] 
[CODE]     
rank description digits who year comment       11538 10^38808010^1124331 388080 CH8 2014 Nearrepdigit (**) 11539 10^38808010^1808681 388080 p377 2014 Nearrepdigit 11540 10^38808010^3329441 388080 p377 2014 Nearrepdigit 11541 10^38808010^3420291 388080 p377 2014 Nearrepdigit 12104 10^37696810^1884841 376968 p404 2018 Nearrepdigit 12949 10^36036010^1830371 360360 p374 2014 Nearrepdigit 18009 10^27720010^990881 277200 p367 2013 Nearrepdigit 18010 10^27720010^1782311 277200 p367 2013 Nearrepdigit 18011 10^27720010^2577681 277200 p372 2013 Nearrepdigit 37645 10^13480910^674041 134809 p235 2010 Nearrepdigit, palindrome 41256 10^10428110^521401 104281 p16 2003 Nearrepdigit, palindrome 45524 10^10000010^614031 100000 p62 2001 Nearrepdigit ...[/CODE] [url]https://primes.utm.edu/primes/search.php[/url] Mathematical Description: ^10^%10^%1 Type: all Maximum number of primes to output: 300 There was an archived project  [url]https://mersenneforum.org/forumdisplay.php?f=107[/url] 
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