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A004023
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Indices of prime repunits: numbers n such that 11...111 (with n 1's) = (10^n - 1)/9 is prime.
(Formerly M2114)
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143
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2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, 5794777, 8177207
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OFFSET
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1,1
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COMMENTS
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People who search for repunit primes or repdigit primes may be looking for this entry.
The indices of primes with digital product (i.e., product of digits) equal to 1.
As of August 2014, only the first five repunits, through (10^1031-1)/9, have been proved prime. The last four repunits are known only to be probable primes and have not been proved to be prime. - Robert Baillie, Aug 17 2014
These indices p must also be prime. If p is not prime, say p = m*n, then 10^(m*n) - 1 = ((10^m)^n) - 1 => 10^m - 1 divides 10^(m*n) - 1. Since 9 divides 10^m - 1 or (10^m - 1)/9 = q, it follows q divides (10^p - 1)/9. This is a result of the identity, a^n - b^n = (a - b)(a^(n-1) + a^(n-2)*b + ... + b^(n-1)). - Cino Hilliard, Dec 23 2008
The numbers R_n = 11...111 = (10^n - 1)/9 with n in this sequence A004023, except for n = 2, are prime repunits in base ten, so they are prime Brazilian numbers belonging to A085104. [See Links: Les nombres brésiliens.] - Bernard Schott, Dec 24 2012
Search limit is 10800000, currently. - Serge Batalov, Jul 01 2021
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REFERENCES
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J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 19, pp 6, Ellipses, Paris 2008.
Graham, Knuth and Patashnik, Concrete Mathematics, Addison-Wesley, 1994; see p 146 problem 22.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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John Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Chris K. Caldwell, The Prime Pages, Top 20: Repunit (lists certified primes with n >= 1000)
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Eric Weisstein's World of Mathematics, Repunit
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EXAMPLE
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2 appears because the 2-digit repunit 11 is prime.
3 does not appear because 111 = 3 * 37 is not prime.
19 appears because the 19-digit repunit 1111111111111111111 is prime.
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MATHEMATICA
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Select[Range[271000], PrimeQ[FromDigits[PadRight[{}, #, 1]]] &] (* Harvey P. Dale, Nov 05 2011 *)
repUnsUpTo[k_] := ParallelMap[If[PrimeQ[#] && PrimeQ[(10^# - 1)/9], #, Nothing] &, Range[k]]; repUnsUpTo[5000] (* Mikk Heidemaa, Apr 24 2017 *)
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PROG
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(PARI) forprime(x=2, 20000, if(ispseudoprime((10^x-1)/9), print1(x", "))) \\ Cino Hilliard, Dec 23 2008
(Magma) [p: p in PrimesUpTo(500) | IsPrime((10^p - 1) div 9)]; // Vincenzo Librandi, Nov 06 2014
(Python) from sympy import isprime; {print(n, end = ', ') for n in range(1, 10**7) if isprime(n) and isprime(10**n//9)} # (Note that sympy.isprime is only a pseudo-primality test.) - Ya-Ping Lu, Dec 20 2021, edited by M. F. Hasler, Mar 28 2022
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CROSSREFS
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KEYWORD
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hard,nonn,nice,more
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AUTHOR
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EXTENSIONS
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a(6) = 49081 PRP found by Harvey Dubner - posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) Sep 09, 1999; proved prime by Paul Underwood, Mar 21 2022.
a(7) = 86453 found using pfgw (a faster version of PrimeForm) on Oct 26 2000 by Lew Baxter (posting to Number Theory List), Oct 26, 2000; proved prime by Andreas Enge, May 16 2023.
a(8) = 109297 was apparently discovered independently by (in alphabetical order) Paul Bourdelais and Harvey Dubner around Mar 26-28 2007.
a(9) = 270343, was found Jul 11 2007 by Maksym Voznyy and Anton Budnyy, subsequently confirmed as a(9) (see Repunit Primes Project link) by Robert Price, Dec 14 2010
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STATUS
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approved
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