Near-repdigit

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.

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:

1. Let one of the digits differ from one--these are the near repunit primes.
2. 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 101888529 - 10944264 - 1 1888529 p423 Oct 2021 Near - repdigit, palindrome 2 993 · 101768283 - 1 1768286 L4879 Feb 2019 Near - repdigit 3 9 · 101762063 - 1 1762064 L4879 Aug 2020 Near - repdigit 4 (10859669 - 1)2 - 2 1719338 p405 May 2022 Near - repdigit 5 8 · 101715905 - 1 1715906 L4879 Aug 2020 Near - repdigit 6 92 · 101585996 - 1 1585998 L4789 Apr 2023 Near - repdigit 7 9992 · 101567410 - 1 1567414 L4879 Aug 2020 Near - repdigit 8 99 · 101536527 - 1 1536529 L4879 Feb 2019 Near - repdigit 9 992 · 101533933 - 1 1533936 L4879 Feb 2019 Near - repdigit 10 92 · 101439761 - 1 1439763 L4789 Dec 2020 Near - repdigit 11 (10657559 - 1)2 - 2 1315118 p405 May 2022 Near - repdigit 12 93 · 101170023 - 1 1170025 L4789 Aug 2022 Near - repdigit 13 2 · 101059002 - 1 1059003 L3432 Sep 2013 Near - repdigit 14 93 · 101029523 - 1 1029525 L4789 Jan 2019 Near - repdigit 15 9 · 101009567 - 1 1009568 L3735 Sep 2016 Near - repdigit 16 96 · 10846519 - 1 846521 L2425 Sep 2011 Near - repdigit 17 92 · 10833852 - 1 833854 L4789 Apr 2018 Near - repdigit 18 (10393063 - 1)2 - 2 786126 p405 May 2022 Near - repdigit 19 (10334568 - 1)2 - 2 669136 p405 May 2022 Near - repdigit 20 93 · 10642225 - 1 642227 L4789 Mar 2020 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