A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.

3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568, 8436308

Offset: 1

Author

Chris K. Caldwell, Jun 22 2005

Keywords

• more
• nonn
• hard

Comments

This allows larger terms of A051254 (which triple in digits each entry) to be given. Like A051254, currently requires Riemann Hypothesis to show sequence continues.

Currently a(11)=66768 generates only a probable prime number. - Arkadiusz Wesolowski, May 28 2011

Likewise a(12) and a(13) generate only a probable prime numbers, as well as being conditional on a(11) and a(12) being proved primes. Minimality of a(12)-a(13) is exhaustively tested. - Serge Batalov, Aug 06 2013

a(14) = 8436308 is found by Ryan Propper and Serge Batalov, Apr 29 2024, but a few remaining gaps below this value were being double-checked. The double-check is now complete (see GitHub link). - Ryan Propper and Serge Batalov, May 24 2024.

Examples

			The Mills' primes (given in A051254) are 2, 2^3+3 = 11, (2^3+3)^3+30 = 11^3+30 = 1361, ((2^3+3)^3+30)^3+6 = 1361^3+6 = 2521008887, etc. The terms added at each step yield this sequence. They are the least positive integers which added to the cube of the preceding prime yield again a prime, cf. formula. - _M. F. Hasler_, Jul 22 2013
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.

Links

• Chris K. Caldwell, Mills' Theorem - a generalization.
• Chris K. Caldwell, The List of Largest Known Primes, The 11th Mills' prime
• Chris K. Caldwell and Yuanyou Cheng, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
• Sergey Batalov, Mills' b(14) search report
• Henri & Renaud Lifchitz, PRP Records and PRP Records, search:Mills
• W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.
• Eric Weisstein's World of Mathematics, Mills' Constant
• Eric Weisstein's World of Mathematics, Mills' Prime
• E. M. Wright, A class of representing functions, J. London Math. Soc., Vol. 29 (1954) pp. 63-71.

Crossrefs

Cf. A051254, A051021.

Programs

Mathematica

B[1] = 2; B[n_] := B[n] = NextPrime[B[n - 1]^3]; Table[B[n + 1] - B[n]^3, {n, 7}] (* Robert Price, Jun 09 2019 *)

PARI

p=2; until(, np=nextprime(p^3); print1(np-p^3, ", "); p=np) \\ Jeppe Stig Nielsen, Apr 22 2020

Formula

b(1) = 2; b(n+1) = nextprime(b(n)^3); a(n) = b(n+1)-b(n)^3;

Extensions

a(9)-a(11) from Caldwell and Cheng, Aug 29 2005

Corrected by T. D. Noe, Sep 24 2008

a(12) (which generates a PRP) from Serge Batalov, Jul 19 2013

a(13) (which generates a PRP) from Serge Batalov, Aug 06 2013

a(14) (which generates a PRP) from Ryan Propper and Serge Batalov, May 24 2024