Factorizations of near-repdigit-related numbers

Table of contents

1. Introduction
2. How to contribute your factors ▶
3. Wanted list ▶
4. Reserved numbers and submitted numbers ▶
5. Contributions ▶
6. Records ▶
7. Graphs ▶
8. News
9. Contributors
10. Factor tables
    • Repunit numbers ▶
    • Near-repdigit numbers of the form AA...AAB ▶
    • Near-repdigit numbers of the form ABB...BB ▶
    • Near-repdigit palindrome numbers of the form AA...AABAA...AA ▶
    • Plateau and depression numbers of the form ABB...BBA ▶
    • Quasi-repdigit numbers of the form ABB...BBC ▶
11. List of near-repdigit-related prime numbers
12. Primality certifications ▶
13. Expression generator of near-repdigit-related numbers ▶
14. Related links ▶

1. Introduction

We are collecting prime factorizations of near-repdigit-related numbers and near-repdigit-related prime numbers.

• Near-repdigit-related numbers
  • Repunit numbers (all digits are the same; 11...11)
  • Near-repdigit numbers (all but one digit are the same; AA...AAB, ABB...BB, etc)
    • Near-repdigit-palindrome numbers (near-repdigit and palindrome; AA...AABAA...AA)
  • Quasi-repdigit numbers (all but two digits are the same; ABB...BBC, etc)
    • Plateau-and-depression numbers (ABB...BBA)
  • others (all but three or more digits are the same)

2. How to contribute your factors ▶

3. Wanted list ▶

4. Reserved numbers and submitted numbers ▶

5. Contributions ▶

6. Records ▶

7. Graphs ▶

8. News

July 6, 2013

Factorizations of 411...117 have been extended up to n=250. Added composite numbers had passed ECM iterations 118 times at the level 35. Most unknown factors are supposed to be greater than 10³⁰.

July 2, 2013

Factorizations of 411...113 have been extended up to n=250. Added composite numbers had passed ECM iterations 118 times at the level 35. Most unknown factors are supposed to be greater than 10³⁰.

July 1, 2013

Borys Jaworski found the first known prime number of the form 7019ｷ10ⁿ-1. Congratulations!

Proven near-repdigit-related primes (except ru/nr/pd/qr)

	digits	flabel	wlabel	expression	when	who
1	881313		70189w	7019ｷ10881309-1	July 1, 2013	Borys Jaworski

• The Prime Database: 7019*10^881309-1 (Chris Caldwell)
• Primes of the form k*10^n-1 (Gary Barnes)
• Riesel conjectures and proofs (Gary Barnes)

News of 2013 ▶

News of 2012 ▶

News of 2011 ▶

News of 2010 ▶

9. Contributors

I am thankful to contributors 10metreh, [AF>Amis des Lapins] Nabz37, [AF>France] intello222222, [AF>HFR>RR] julien76100, [AF>Le_Pommier>MacADSL.com]Bertrand, [AF>Le_Pommier>MacBidouille.com]m.o.u.s.t.i.c, [boinc.at] Fireman69, [GPU Force] Robert 7NBI, [P3D] Crashtest, [SG-FC] hl, [SG]marodeur6, [TiDC] Chulma, [XTBA>TSA] IvanleFou, [XTBA>TSA]+IvanleFou, Agnew yoyo, Alessandro Freda, Alexander Mkrtychyan, Alfred Reich, Andreas Tete, Andy, Anonymous, Anton Korobeynikov, apo, axels, Bart Jans, bbd in ep, bbmz, Beyond, Bob Price, Bouk de Water, Brad, Bryan Koen, btolksdorf, bundaboy, Carlos Pinho, Cedric Vonck, centylion torun, Chris Monico, Christopher Birkbeck, Crunchers For More Power, Crystal Pellet, Darren Bedwell, David Broadhurst, denjoR, Dennis Bertels, Dmitry Domanov, Domanov Dmitry, Dubslow, Dull Man Real Dull, Edwin Hall, Egon Olsen, EMK2, ERBrouwer, Erik Branger, Familie, Fragman, Frank Schickel, Frank Villasenor, frank_0987923, Freyn, Friedhelm Baumeister, G.L.I.S., Gennady Stolyarov II, GRB IV, Greg Childers, Grubix, Grzegorz Roman Granowski, Ha Seok woo, Harvey Dubner, honeycrack7, Hugo Platzer, Ian, Ignacio Santos, Jamoko, Jan Ptacek, Jarekcz, jdommer, Jeff Gilchrist, Jeff17, Jens, Jerzy Gieorgijewski, JMB, Jo Yeong Uk, John Black, Joshua2, JPascoa, Julien Peter Benney, juno1369, Justin Card, K, Maemondo, Kenichiro Yamaguchi, Kenji Ibusuki, kh9, KTakahashi, Lazarusuk, Lionel Debroux, Liuqyn, Luca Dentis, Luigi Morelli, Makoto Kamada, Makso, Maksym Voznyy, MarcinGorecki, Markus Tervooren, Mathew, matsui, Max Dettweiler, Meharts, mia7077, Michael Peterson, Michael Rao, MicroKraken, mikkovi, Mitchell, Mr. Hankey, Mr. Ice, msc_nbg, Naoki Yamamoto, Nechaev Sergey, nenym, NFS@Home, Nicolai Caraibendko, nisba, Nitro, Norbert Schneider, Patrick Keller, Pawel Apostol, Phil, Phil Carmody, Philippe Strohl, Pipao, Polybius, processing-home, pschoefer, Raffi Chaglassian, Ray Chandler, RAZIELakaALIN, Rich Dickerson, RichD, rkillian, Roald, Robert Backstrom, RobertS, Rocke Verser, RSALS, ruffenach timothee, Samuel Chong, Sander Hoogendoorn, SBS, Schildkroete, Scott O, Sean A. Irvine, Serge Batalov, Shaopu Lin, shauge, shima@engei, Shusuke Kubota, shyguy7129, Sinkiti Sibata, skneo, Sonic@Planet 3DNow!, steinrar, stelf, Steph Lady, suberi, Takahiro Nohara, Tapio Rajala, Tetsuya Kobayashi, Thomas Womack, Tomoya Adachi, toms83, ToolboxNL, tor, transtec-AG, Tyler Cadigan, UA_ReMMeR, vanos0512, VictordeHollander, Vladimir, Warut Roonguthai, Wataru Sakai, Wojciech Florek, Yang Hae Hun, Yoichi Hanatani, yoshida, Youcef L, Youcef Lemsafer, yoyo, Zeta-Flux and zlodeck.

10. Factor tables

Repunit numbers ▶

Near-repdigit numbers of the form AA...AAB ▶

Near-repdigit numbers of the form ABB...BB ▶

Near-repdigit palindrome numbers of the form AA...AABAA...AA ▶

Plateau and depression numbers of the form ABB...BBA ▶

Quasi-repdigit numbers of the form ABB...BBC ▶

11. List of near-repdigit-related prime numbers

• primesize.txt (primesize.zip) sorted by size, 100 digits or more
  Example 1: "88881 8w1 (8*10^27811-71)/9 -33.00%_7334" means that the ratio of the factored part of (8ｷ10²⁷⁸¹¹-71)/9-1 is 33.00% and the largest known (probable) prime factor of (8ｷ10²⁷⁸¹¹-71)/9-1 is a 7334-digit number. In other words, to prove the primality of (8ｷ10²⁷⁸¹¹-71)/9, you have to prove primality of the 7334-digit number by ECPP and use N-1/N+1 test with Konyagin-Pomerance extension.
  Command line to find PRPs for N-1/N+1/KP/CHG:
  grep "\(2[5-9]\|[3-9][0-9]\|100\)\.[0-9][0-9]%" primesize.txt
  Example 2: "71111 71w (64*10^83461-1)/9 -1:1:2:2" means that (64ｷ10⁸³⁴⁶¹-1)/9-1 is factored algebraically into four parts. In fact, (64ｷ10⁸³⁴⁶¹-1)/9-1 = 10ｷ(2ｷ10¹³⁹¹⁰+1)/3ｷ(2ｷ10¹³⁹¹⁰-1)ｷ(4ｷ10²⁷⁸²⁰-2ｷ10¹³⁹¹⁰+1)/3ｷ(4ｷ10²⁷⁸²⁰+2ｷ10¹³⁹¹⁰+1). Such numbers may be easily factored than other numbers.
  Example 3: "37773 37w3 (34*10^15768-43)/9 +proven@7884" means that the primality of (34ｷ10¹⁵⁷⁶⁸-43)/9 was proved by N-1/N+1 test with ECPP certification of a prime cofactor of Phi₇₈₈₄(10). See Primality certifications page.
• primecount.txt, primecount2.txt and primecount3.txt sorted by count
• primedifficulty.txt sorted by difficulty

12. Primality certifications ▶

13. Expression generator of near-repdigit-related numbers ▶

14. Related links ▶