Primes made mostly of nines

[fivemack]

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 ?

[xilman]

Originally Posted by fivemack

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 ?

Why do you care?

It isn't that 10k-digit numbers are difficult to prove prime by ECPP or APR-CL these days.

[kruoli]

FactorDB instantly proved it by N+1 as being prime.

[R. Gerbicz]

Originally Posted by fivemack

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 ?

Combined Theorem 1 is enough from https://primes.utm.edu/prove/prove3_3.html
with F1=1, F2=10^8668.

[fivemack]

Originally Posted by xilman

Why do you care?

It isn't that 10k-digit numbers are difficult to prove prime by ECPP or APR-CL these days.

Using a gigahertz-month 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 6400-digit prime comprised entirely of nines with a single eight, and I thought this was not a particularly exciting result)

[paulunderwood]

Originally Posted by fivemack

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 ?

./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, Brillhart-Lehmer-Selfridge]                                    
Running N+1 test using discriminant 43, base 1+sqrt(43)                                    
10^10000 - 10^8668 - 1 is prime! (10.6651s+0.0255s)

Back in the day, we found this one 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.

cat NRD_gigantic 
ABC2 10^10000-10^$a-1
a: from 1 to 9999

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.
                                    
10^10000-10^750-1 has factors: 2313617                                                                         
10^10000-10^1589-1 has factors: 2635553                                                                         
10^10000-10^3486-1 is 3-PRP! (1.1229s+0.0885s)                                                                  
10^10000-10^3909-1 is 3-PRP! (1.0102s+0.1867s)                                                                  
10^10000-10^4151-1 has factors: 376769                                                                          
10^10000-10^5133-1 is 3-PRP! (1.0614s+0.0897s)                                                                  
10^10000-10^5334-1 has factors: 772147                                                                          
10^10000-10^6134-1 has factors: 2749921                                                                         
10^10000-10^7928-1 is 3-PRP! (1.1574s+0.1369s)                                                                  
10^10000-10^8072-1 has factors: 2742227                                                                         
10^10000-10^8668-1 is 3-PRP! (0.9757s+0.0931s)                                                                  
10^10000-10^8740-1 has factors: 2600837                                                                         
                                                                                                                

real	34m58.010s
user	34m57.090s
sys	0m0.524s

./pfgw64 -tp -q"10^10000-10^3486-1"
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 10^10000-10^3486-1 [N+1, Brillhart-Lehmer-Selfridge]                                    
Running N+1 test using discriminant 7, base 1+sqrt(7)                                    
10^10000-10^3486-1 is prime! (3.8788s+0.0002s)                                                  

./pfgw64 -tp -q"10^10000-10^3909-1"
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 10^10000-10^3909-1 [N+1, Brillhart-Lehmer-Selfridge]                                    
Running N+1 test using discriminant 7, base 1+sqrt(7)                                    
10^10000-10^3909-1 is prime! (3.8218s+0.0001s)                                                  

./pfgw64 -tp -q"10^10000-10^5133-1"
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 10^10000-10^5133-1 [N+1, Brillhart-Lehmer-Selfridge]                                    
Running N+1 test using discriminant 7, base 1+sqrt(7)                                    
10^10000-10^5133-1 is prime! (3.9534s+0.0001s)                                                  

./pfgw64 -tp -q"10^10000-10^7928-1"
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 10^10000-10^7928-1 [N+1, Brillhart-Lehmer-Selfridge]                                    
Running N+1 test using discriminant 7, base 1+sqrt(7)                                    
10^10000-10^7928-1 is prime! (4.5040s+0.0002s)

[Batalov]

-----  -------------------------------- ------- ----- ---- --------------
 rank           description              digits  who year comment
-----  -------------------------------- ------- ----- ---- --------------
11538  10^388080-10^112433-1             388080   CH8 2014 Near-repdigit (**)
11539  10^388080-10^180868-1             388080  p377 2014 Near-repdigit
11540  10^388080-10^332944-1             388080  p377 2014 Near-repdigit
11541  10^388080-10^342029-1             388080  p377 2014 Near-repdigit
12104  10^376968-10^188484-1             376968  p404 2018 Near-repdigit
12949  10^360360-10^183037-1             360360  p374 2014 Near-repdigit
18009  10^277200-10^99088-1              277200  p367 2013 Near-repdigit
18010  10^277200-10^178231-1             277200  p367 2013 Near-repdigit
18011  10^277200-10^257768-1             277200  p372 2013 Near-repdigit
37645  10^134809-10^67404-1              134809  p235 2010 Near-repdigit, palindrome
41256  10^104281-10^52140-1              104281   p16 2003 Near-repdigit, palindrome
45524  10^100000-10^61403-1              100000   p62 2001 Near-repdigit
...

https://primes.utm.edu/primes/search.php
Mathematical Description: ^10^%-10^%-1
Type: all
Maximum number of primes to output: 300

There was an archived project - https://mersenneforum.org/forumdisplay.php?f=107