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 least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006, corrected by M. F. Hasler, Sep 11 2024
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.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.13, p. 130.
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
(PARI) apply( {A051254(n, p=2)=while(n--, p=nextprime(p^3)); p}, [1..6]) \\ M. F. Hasler, Sep 11 2024
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, May 05 2007
STATUS
approved