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 96 · 10⁸⁴⁶⁵¹⁹ - 1 846521 L2425 Sep 2011 Near - repdigit
2 8 · 10⁶⁰⁸⁹⁸⁹ - 1 608990 p297 May 2011 Near - repdigit
3 5 · 10⁵¹¹⁰⁵⁶ - 1 511057 p297 Mar 2011 Near - repdigit
4 5 · 10⁴⁶⁴⁸⁴³ - 1 464844 p297 Feb 2011 Near - repdigit
5 5 · 10⁴⁴⁵⁷⁷³ - 1 445774 p297 Jan 2011 Near - repdigit
6 6 · 10⁴¹⁴⁵⁰⁸ - 1 414509 p297 Jan 2011 Near - repdigit
7 9 · 10³⁸³⁶⁴³ - 1 383644 p297 Jan 2011 Near - repdigit
8 9 · 10³⁶⁴⁵²¹ - 1 364522 p297 Dec 2010 Near - repdigit
9 9999992 · 10³⁶⁰⁴⁰³ - 1 360410 L1958 Nov 2011 Near - repdigit
10 9999993 · 10³³⁵⁹⁰⁵ - 1 335912 L1958 Apr 2013 Near - repdigit
11 9999993 · 10³³¹⁹³⁸ - 1 331945 L1958 Apr 2013 Near - repdigit
12 6 · 10³¹⁹⁸⁸⁹ - 1 319890 p297 Nov 2010 Near - repdigit
13 10³¹⁴⁷²⁷ - 8 · 10¹⁵⁷³⁶³ - 1 314727 p235 Jan 2013 Near - repdigit, palindrome
14 992 · 10³¹²¹³⁶ - 1 312139 p356 Apr 2013 Near - repdigit
15 6 · 10²⁹³¹³⁴ - 1 293135 p297 Nov 2010 Near - repdigit
16 10²⁹⁰²⁵³ - 2 · 10¹⁴⁵¹²⁶ - 1 290253 p235 Apr 2012 Near - repdigit, Palindrome
17 6 · 10²⁷⁰⁶⁵⁸ - 1 270659 p297 Nov 2010 Near - repdigit
18 10²⁶⁹⁴⁷⁹ - 7 · 10¹³⁴⁷³⁹ - 1 269479 p235 Feb 2012 Near - repdigit, Palindrome
19 6 · 10²⁶⁷⁵⁹⁸ - 1 267599 p297 Nov 2010 Near - repdigit
20 9 · 10²⁶⁰²⁵³ - 1 260254 p297 Nov 2010 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