A051254 Mills primes.

2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499

Offset

1,1

Comments

Mills showed that there is a number A > 1 but not an integer, such that floor( A^(3^n) ) is a prime for all n = 1, 2, 3, ... A is approximately 1.306377883863... (see A051021).

a(1) = 2 and (for n > 1) a(n) is greatest prime < a(n-1)^3. - Jonathan Vos Post, May 05 2006

The name refers to the American mathematician William Harold Mills (1921-2007). - Amiram Eldar, Jun 23 2021

References

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

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..8

Chris K. Caldwell, Mills' Theorem - a generalization.

Chris K. Caldwell and Yuanyou Chen, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.

Steven R. Finch, Mills' Constant. [Broken link]

Steven R. Finch, Mills' Constant. [From the Wayback machine]

Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, Amer. Math. Monthly, Vol. 126, No. 1 (2019), pp. 72-73; ResearchGate link, arXiv preprint, arXiv:2010.15882 [math.NT], 2020.

James Grime and Brady Haran, Awesome Prime Number Constant, Numberphile video, 2013.

Brian Hayes, Pumping the Primes, bit-player, Aug 19 2015.

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

William H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53, No. 6 (1947), p. 604; Errata, ibid., Vol. 53, No 12 (1947), p. 1196.

Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.

László Tóth, A Variation on Mills-Like Prime-Representing Functions, arXiv:1801.08014 [math.NT], 2018.

Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.

Eric Weisstein's World of Mathematics, Mills' Prime.

Eric Weisstein's World of Mathematics, Prime Formulas.

Eric W. Weisstein, Table of n, a(n) for n = 1..13.

Formula

a(1) = 2; a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006

Example

a(3) = 1361 = 11^3 + 30 = a(2)^3 + 30 and there is no smaller k such that a(2)^3 + k is prime. - Jonathan Vos Post, May 05 2006

Maple

floor(A^(3^n), n=1..10); # A is Mills's constant: 1.306377883863080690468614492602605712916784585156713644368053759966434.. (A051021).

Mathematica

p = 1; Table[p = NextPrime[p^3], {6}] (* T. D. Noe, Sep 24 2008 *)
NestList[NextPrime[#^3] &, 2, 5] (* Harvey P. Dale, Feb 28 2012 *)

Prog

(PARI) a(n)=if(n==1, 2, nextprime(a(n-1)^3)) \\ Charles R Greathouse IV, Jun 23 2017

Crossrefs

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908, A118909, A118910, A118911, A118912, A118913.

Cf. A224845 (integer lengths of Mills primes).

Cf. A108739 (sequence of offsets b_n associated with Mills primes).

Cf. A051021 (decimal expansion of Mills constant).

Sequence in context: A062636 A343900 A343929 * A095820 A101295 A131306

Adjacent sequences: A051251 A051252 A051253 * A051255 A051256 A051257

Keyword

nonn,nice

Author

Simon Plouffe.

Extensions

Edited by N. J. A. Sloane, May 05 2007

Status

approved