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

2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, 784, 1732, 1918, 8855, 11245, 11960, 12130, 18533, 22718, 23365, 24253, 24549, 25324, 30178, 53718, 380976, 424861, 563535, 666903

Offset: 1

Author

Labos Elemer, Jul 10 2000

Keywords

• nonn

Comments

Also numbers k such that (10^k-7)/3 is prime.

Sierpiński attributes the primes for k = 2,...,8 to A. Makowski.

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.)

Concerning certifying primes, see the references by Goldwasser et al., Atkin et al. and Morain. - Labos

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

(10^(15m+2)-7)/3 == 0 (mod 31). So 15m+2 isn't a term for m > 0. - Seiichi Manyama, Nov 05 2016

References

• C. Caldwell, The near repdigit primes 333...331, J.Recreational Math. 21:4 (1989) 299-304.
• S. Goldwasser and J. Kilian, Almost All Primes Can Be Quickly Certified. in Proc. 18th STOC, 1986, pp. 316-329.
• W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.

Links

• A. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp. 61:29-68, 1993.
• Makoto Kamada, Prime numbers of the form 33...331.
• Mathematics.StackExchange.com, 31,331,3331, 33331,333331,3333331,33333331 are prime
• F. Morain, Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm, INRIA Research Report, # 911, October 1988.
• Dave Rusin, Primes in exponential sequences [Broken link]
• Dave Rusin, Primes in exponential sequences [Cached copy]
• Index entries for primes involving repunits

Crossrefs

Cf. A051200, A033175, A055520.

Programs

Mathematica

Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
One may run the prime certificate program as follows <True]}, {n, 1, 16}] (* Labos Elemer *)

PARI

for(n=1,2000, if(isprime((10^n-7)/3),print(n)))

Formula

a(n) = A055520(n) + 1.

Extensions

Corrected and extended by Jason Earls, Sep 22 2001

a(20)-a(31) were found by Makoto Kamada (see links for details). At present they correspond only to probable primes.

a(32)-a(33) from Leonid Durman, Jan 09-10 2012

a(34)-a(35) from Kamada data by Tyler Busby, Apr 14 2024