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%I M2114 #248 Aug 05 2023 23:25:15
%S 2,19,23,317,1031,49081,86453,109297,270343,5794777,8177207
%N Indices of prime repunits: numbers n such that 11...111 (with n 1's) = (10^n - 1)/9 is prime.
%C People who search for repunit primes or repdigit primes may be looking for this entry.
%C The indices of primes with digital product (i.e., product of digits) equal to 1.
%C 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
%C 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
%C 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
%C Search limit is 10800000, currently. - _Serge Batalov_, Jul 01 2021
%D J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 19, pp 6, Ellipses, Paris 2008.
%D Graham, Knuth and Patashnik, Concrete Mathematics, Addison-Wesley, 1994; see p 146 problem 22.
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Paul Bourdelais, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906">A Generalized Repunit Conjecture</a>, NMBRTHRY, 25 Jun 2009.
%H John Brillhart, <a href="/A001562/a001562.pdf">Letter to N. J. A. Sloane, Aug 08 1970</a>
%H John Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
%H Chris K. Caldwell, The Prime Pages, <a href="https://t5k.org/top20/page.php?id=57">Top 20: Repunit</a> (lists certified primes with n >= 1000)
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/circular.htm">Circular Primes</a>
%H Giovanni Di Maria, <a href="http://www.elektrosoft.it/matematica/repunit/repunit.htm">Repunit Primes Project</a>
%H Harvey Dubner, <a href="/A028491/a028491.pdf">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
%H Harvey Dubner, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;1d525256.9909">New probable prime repunit, R(49081)</a>, Number Theory List, Sep 09 1999.
%H Harvey Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-01-01319-9">Repunit R49081 is a probable prime</a>, Math. Comp., 71 (2001), 833-835.
%H Harvey Dubner, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;a9325f9e.0704">Posting to Number Theory List : Apr 03 2007</a>
%H Martianus Frederic Ezerman, Bertrand Meyer, and Patrick Solé, <a href="http://arxiv.org/abs/1210.7593">On Polynomial Pairs of Integers</a>, arXiv:1210.7593 [math.NT], 2012-2014.
%H Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Ezerman/eze3.html">On Polynomial Pairs of Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit">Factorizations of 11...11 (Repunit)</a>.
%H Henri Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Mersenne and Fermat primes field</a>
%H T. Muller, <a href="https://doi.org/10.4171/EM/183">Ist die Folge der Primzahl-quersummen beschrankt?</a>, Elem. Math. 66 (2011) 146-154; doi:10.4171/EM/183.
%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
%H Andy Steward, <a href="http://www.users.globalnet.co.uk/~aads/primes.html">Prime Generalized Repunits</a>
%H Sam Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>
%H E. Wegrzynowski, <a href="http://www.lifl.fr/~wegrzyno/RepunitBase10/repbase101.html">Nombres 1_[n] premiers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RepunitPrime.html">Repunit Prime</a>
%H H. C. Williams and Harvey Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1986-0856714-3">The primality of R1031</a>, Math. Comp., 47(176), Oct 1986, 703-711.
%H <a href="/index/Pri#Pri_rep">Index entries for primes involving repunits</a>
%e 2 appears because the 2-digit repunit 11 is prime.
%e 3 does not appear because 111 = 3 * 37 is not prime.
%e 19 appears because the 19-digit repunit 1111111111111111111 is prime.
%t Select[Range[271000], PrimeQ[FromDigits[PadRight[{}, #, 1]]] &] (* _Harvey P. Dale_, Nov 05 2011 *)
%t repUnsUpTo[k_] := ParallelMap[If[PrimeQ[#] && PrimeQ[(10^# - 1)/9], #, Nothing] &, Range[k]]; repUnsUpTo[5000] (* _Mikk Heidemaa_, Apr 24 2017 *)
%o (PARI) forprime(x=2,20000,if(ispseudoprime((10^x-1)/9),print1(x","))) \\ _Cino Hilliard_, Dec 23 2008
%o (Magma) [p: p in PrimesUpTo(500) | IsPrime((10^p - 1) div 9)]; // _Vincenzo Librandi_, Nov 06 2014
%o (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
%Y See A004022 for the actual primes.
%Y Cf. A055557, A002275, A085104.
%K hard,nonn,nice,more
%O 1,1
%A _N. J. A. Sloane_
%E 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.
%E 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.
%E a(8) = 109297 was apparently discovered independently by (in alphabetical order) _Paul Bourdelais_ and Harvey Dubner around Mar 26-28 2007.
%E 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
%E a(10) = 5794777 was found Apr 20 2021 by _Ryan Propper_ and _Serge Batalov_
%E a(11) = 8177207 was found May 08 2021 by _Ryan Propper_ and _Serge Batalov_
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