Near-repdigit

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The Prime Pages

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

Definitions and Notes

A repunit is a number of the form 11111...111 (repeated units). In base two (binary), these are the Mersenne primes. In base ten, just a few are known. If we repeat any other digit, then we get a composite (e.g., 777777 is divisible by 7).

To get a more general form, two things have been tried:

Let one of the digits differ from one--these are the near repunit primes.
Let all but one of the digits be the same, these are the near repdigit primes (and include the near repunit primes).

Record Primes of this Type

rank prime digits who when comment

1 2 · 10^1059002 - 1 1059003 L3432 Sep 2013 Near - repdigit
2 9 · 10^1009567 - 1 1009568 L3735 Sep 2016 Near - repdigit
3 96 · 10⁸⁴⁶⁵¹⁹ - 1 846521 L2425 Sep 2011 Near - repdigit
4 92 · 10⁸³³⁸⁵² - 1 833854 L4789 Apr 2018 Near - repdigit
5 8 · 10⁶⁰⁸⁹⁸⁹ - 1 608990 p297 May 2011 Near - repdigit
6 92 · 10⁵⁴⁴⁹⁰⁵ - 1 544907 L3735 May 2015 Near - repdigit
7 5 · 10⁵¹¹⁰⁵⁶ - 1 511057 p297 Mar 2011 Near - repdigit
8 95 · 10⁴⁶⁶⁰⁰² - 1 466004 L3735 May 2014 Near - repdigit
9 5 · 10⁴⁶⁴⁸⁴³ - 1 464844 p297 Feb 2011 Near - repdigit
10 5 · 10⁴⁴⁵⁷⁷³ - 1 445774 p297 Jan 2011 Near - repdigit
11 6 · 10⁴¹⁴⁵⁰⁸ - 1 414509 p297 Jan 2011 Near - repdigit
12 99998 · 10³⁸⁹¹⁵⁰ - 1 389155 L3432 Sep 2016 Near - repdigit
13 10³⁸⁸⁰⁸⁰ - 10¹¹²⁴³³ - 1 388080 CH8 Nov 2014 Near - repdigit
14 10³⁸⁸⁰⁸⁰ - 10¹⁸⁰⁸⁶⁸ - 1 388080 p377 Nov 2014 Near - repdigit
15 10³⁸⁸⁰⁸⁰ - 10³³²⁹⁴⁴ - 1 388080 p377 Dec 2014 Near - repdigit
16 10³⁸⁸⁰⁸⁰ - 10³⁴²⁰²⁹ - 1 388080 p377 Dec 2014 Near - repdigit
17 99999995 · 10³⁸⁶⁹⁵⁶ - 1 386964 L3432 Sep 2016 Near - repdigit
18 9 · 10³⁸³⁶⁴³ - 1 383644 p297 Jan 2011 Near - repdigit
19 10³⁷⁶⁹⁶⁸ - 10¹⁸⁸⁴⁸⁴ - 1 376968 p404 Sep 2018 Near - repdigit
20 9999998 · 10³⁶⁹⁷⁰⁵ - 1 369712 L1958 Jan 2014 Near - repdigit

References

Caldwell89
C. Caldwell, "The near repdigit primes 333 ^... 331," J. Recreational Math., 21:4 (1989) 299--304.
Caldwell90
C. Caldwell, "The near repdigit primes A_nB, AB_n, and UBASIC," J. Recreational Math., 22:2 (1990) 100--109.
CD95
C. Caldwell and H. Dubner, "The near repunit primes 1_n-k-10₁1_k," J. Recreational Math., 27 (1995) 35--41.
CD97
C. Caldwell and H. Dubner, "The near repdigit primes A_n-k-1B₁A_k, especially 9_n-k-18₁9_k," J. Recreational Math., 28:1 (1996-97) 1--9.
Heleen98
Heleen, J. P., "More near-repunit primes 1_n-k-1D₁1_k, D=2,3, ..., 9," J. Recreational Math., 29:3 (1998) 190--195.
Williams78b
H. C. Williams, "Some primes with interesting digit patterns," Math. Comp., 32 (1978) 1306--1310. Corrigendum in 39 (1982), 759. MR 58:484