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A055557 Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

%I A055557 #53 Apr 14 2024 02:35:10
%S A055557 2,3,4,5,6,7,8,18,40,50,60,78,101,151,319,382,784,1732,1918,8855,
%T A055557 11245,11960,12130,18533,22718,23365,24253,24549,25324,30178,53718,
%U A055557 380976,424861,563535,666903
%N A055557 Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.
%C A055557 Also numbers k such that (10^k-7)/3 is prime.
%C A055557 Sierpiński attributes the primes for k = 2,...,8 to A. Makowski.
%C A055557 The history of the discovery of these numbers may be as follows: a(1)-a(7), Makowski; a(8)-a(18), Caldwell; a(19), Earls; a(20)-a(31), Kamada. (Corrections to this account will be welcomed.)
%C A055557 Concerning certifying primes, see the references by Goldwasser et al., Atkin et al. and Morain. - Labos
%C A055557 No more than 14 consecutive exponents can provide primes because for exponents 15m+2, 16m+9, 18m+12, 22m+21, terms are divisible by 31, 17, 19, 23 respectively. Here 7 of possible 14 is realized. - _Labos Elemer_, Jan 19 2005
%C A055557 (10^(15m+2)-7)/3 == 0 (mod 31). So 15m+2 isn't a term for m > 0. - _Seiichi Manyama_, Nov 05 2016
%D A055557 C. Caldwell, The near repdigit primes 333...331, J.Recreational Math. 21:4 (1989) 299-304.
%D A055557 S. Goldwasser and J. Kilian, Almost All Primes Can Be Quickly Certified. in Proc. 18th STOC, 1986, pp. 316-329.
%D A055557 W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.
%H A055557 A. Atkin and F. Morain, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1199989-X">Elliptic Curves and Primality Proving</a>, Math. Comp. 61:29-68, 1993.
%H A055557 Makoto Kamada, <a href="https://stdkmd.net/nrr/3/33331.htm#prime">Prime numbers of the form 33...331</a>.
%H A055557 Mathematics.StackExchange.com, <a href="http://math.stackexchange.com/questions/542634/31-331-3331-33331-333331-3333331-33333331-are-prime">31,331,3331, 33331,333331,3333331,33333331 are prime</a>
%H A055557 F. Morain, <a href="https://hal.inria.fr/inria-00075645">Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm</a>, INRIA Research Report, # 911, October 1988.
%H A055557 Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/98/exp_primes">Primes in exponential sequences</a> [Broken link]
%H A055557 Dave Rusin, <a href="/A055557/a055557.txt">Primes in exponential sequences</a> [Cached copy]
%H A055557 <a href="/index/Pri#Pri_rep">Index entries for primes involving repunits</a>
%F A055557 a(n) = A055520(n) + 1.
%t A055557 Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
%t A055557 One may run the prime certificate program as follows <<NumberTheory`PrimeQ` Table[{n, ProvablePrimeQ[(-7+10^Part[t, n])/3, Certificate->True]}, {n, 1, 16}] (* _Labos Elemer_ *)
%o A055557 (PARI) for(n=1,2000, if(isprime((10^n-7)/3),print(n)))
%Y A055557 Cf. A051200, A033175, A055520.
%K A055557 nonn
%O A055557 1,1
%A A055557 _Labos Elemer_, Jul 10 2000
%E A055557 Corrected and extended by _Jason Earls_, Sep 22 2001
%E A055557 a(20)-a(31) were found by Makoto Kamada (see links for details). At present they correspond only to probable primes.
%E A055557 a(32)-a(33) from Leonid Durman, Jan 09-10 2012
%E A055557 a(34)-a(35) from Kamada data by _Tyler Busby_, Apr 14 2024