mersenneforum.org  

Go Back   mersenneforum.org > Prime Search Projects > And now for something completely different

Reply
 
Thread Tools
Old 2014-09-22, 18:39   #1
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22·2,281 Posts
Default Near- and quasi-repunit PRPs

M.Kamada has been collecting the near- and quasi-repunit primes and PRPs for many years.

I've recently revisited this topic and remembered a few pitfalls that new searchers might fall in, so I decided to list them here:
1. Sieving is possible with sr*sieve, though a few tips are needed. First, because most forms have an implied denominator of 3 or 9, one needs to not sieve for p=3. For most forms srsieve works (with -p 5); start with srsieve to low limit, then use awk or similar to remove the 3-based composites. Then proceed with sr{1|2}sieve if possible (e.g. for +/-1 forms.
2. PRP testing is as fast as "normal" numbers (without denominator). For that, either use Prime95 (with worktodo entries like PRP={k},{b},{n},{c},"denominator") or LLR with ABC($a*$b^$c$d)/$e header! LLR runs a nice battery test: both SPRP and then a strong-Fermat, Lucas and Frobenius.
3. Use PFGW only for independent validation, because it uses general FFT and is many times slower.

Example of application (711...11 series):
1. Find current limits at Kamada's page.
2. Sieve with srsieve ... -p 5 -P 2e6 "64*10^n-1"
3. Remove all entries where n != 1 (mod 6). This is because even n's (0,2,4) generate diff.of squares; n=3 (mod 6) generate diff.of cubes; and n=5 (mod 6) are divisible by additional 3's after being divided by 9.
4. Sieve with sr1sieve (or combine with some other siblings, e.g. I did the 133..33 series, and sieve with sr2sieve)
5. Prepare P95 or LLR input files with awk and run them.
6. ??????
7. PROFIT! (64·10^762811-1)/9 is a (S-F-L-F)PRP.

Similarly, (64·10^779465-1)/81 is a (non-repunit) SPRP, and (89·10^411590+1)/9 is a Plateau-and-Depression SPRP ("988...889").

All these are still in the PRP Top processing queue.
Batalov is offline   Reply With Quote
Old 2014-09-22, 19:35   #2
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22·2,281 Posts
Smile

P.S. We can imaginatively call the (64·10^779465-1)/81 a "rainbow" number, because it has all decimal digits (except 8) in order, over and over again.
Code:
79012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679012345679...
Batalov is offline   Reply With Quote
Old 2014-10-25, 03:25   #3
davar55
 
davar55's Avatar
 
May 2004
New York City

2×32×5×47 Posts
Default

Quote:
Originally Posted by Batalov View Post
P.S. We can imaginatively call the (64·10^779465-1)/81 a "rainbow" number, because it has all decimal digits (except 8) in order, over and over again.
nice name

davar55 is offline   Reply With Quote
Old 2014-10-27, 11:03   #4
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22·2,281 Posts
Wink A curio: "5" followed by 51111 "1"s is a PRP

Here is a small but peculiar PRP. (self-describing, sort of)
It is from the "511..11" near-repunit series, and in decimal expansion it has a "5" followed by 51111 "ones".

Code:
(46*10^51111-1)/9 is strong-Fermat, Lucas and Frobenius PRP! (P = 5, Q = 3, D = 13)  Time : 88.194 sec.
Batalov is offline   Reply With Quote
Old 2014-10-31, 00:29   #5
pepi37
 
pepi37's Avatar
 
Dec 2011
After milion nines:)

24758 Posts
Default

(91*10^112098-1)/9

One 1 then one 0 then 1 until end :)

Thanks Batalov for showing me way for sieving and llr-ing this type of PRP.

Last fiddled with by pepi37 on 2014-10-31 at 00:30 Reason: edit PRP
pepi37 is online now   Reply With Quote
Old 2014-11-03, 22:28   #6
pepi37
 
pepi37's Avatar
 
Dec 2011
After milion nines:)

32×149 Posts
Default

Quote:
Originally Posted by pepi37 View Post
(91*10^112098-1)/9

One 1 then one 0 then 1 until end :)

Thanks Batalov for showing me way for sieving and llr-ing this type of PRP.
And new one: double in size :)
(73*10^248145-1)/9
pepi37 is online now   Reply With Quote
Old 2019-06-04, 19:27   #7
sweety439
 
sweety439's Avatar
 
Nov 2016

23×97 Posts
Default

Found A200065(1033): (10^3376-1)/9-78 is prime!!!
sweety439 is offline   Reply With Quote
Old 2019-06-04, 20:08   #8
sweety439
 
sweety439's Avatar
 
Nov 2016

23·97 Posts
Default

Quote:
Originally Posted by sweety439 View Post
Found A200065(1033): (10^3376-1)/9-78 is prime!!!
Another term is A200065(1079): (10^2553-1)/9-32 is prime!!!
sweety439 is offline   Reply With Quote
Old 2019-06-04, 20:11   #9
sweety439
 
sweety439's Avatar
 
Nov 2016

8B716 Posts
Default

Note that A200065(1073)=0, since (10^n-1)/9-38 is divisible by either 3 or 37 for n not divisible by 3, and has algebra factors (divisible by (10^(n/3)-7)/3, this number is >1 for n>3) for n divisible by 3.

Last fiddled with by sweety439 on 2019-06-04 at 20:14
sweety439 is offline   Reply With Quote
Old 2019-09-12, 13:06   #10
Dylan14
 
Dylan14's Avatar
 
"Dylan"
Mar 2017

29 Posts
Default

I'm looking to work on the 31111 series, and I want to make sure I have the procedure right before I start:


1. Since 31111 = 31w = (28*10^n-1)/9, there is no need to sieve p=3, and so we start with p = 5. Run srsieve to low limits (say, to 1e7) with the --newpgen flag, starting with n = 200001 (as it has been searched to 200000 from Kamada's website) to whatever max n (say, 500000).
2. Because by 2.4.1 of https://stdkmd.net/nrr/3/31111.htm, there is no need to test any n where n mod 3 = 0, so we can remove them. Not sure if it be worth removing the other factors shown in that section.
3. Continue sieve with sr1sieve up to desired sieve limit
4. Change the file header to an ABC header using srfile, and then add /9 at the end, so it would read
Code:
ABC (28*10^$a-1)/9
5. Use LLR to test and (hopefully) find PRP's.


Let me know.
Dylan14 is offline   Reply With Quote
Old 2019-09-12, 13:31   #11
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22×2,281 Posts
Default

ABC ($a*$b^$c$d)/$e
28 10 n -1 9
Batalov is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Searching for generalized repunit PRP sweety439 sweety439 227 2020-08-18 01:46
Generalized Repunit primes Bob Underwood Math 11 2017-01-25 11:19
generalized repunit PRP sweety439 And now for something completely different 1 2016-12-07 15:58
PRPs not prime schickel FactorDB 1 2015-08-03 02:50
Proven PRPs? Random Poster FactorDB 0 2012-07-24 10:53

All times are UTC. The time now is 07:34.

Sun Sep 20 07:34:17 UTC 2020 up 10 days, 4:45, 0 users, load averages: 1.35, 1.26, 1.27

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.