Mills’ constant

Find the smallest real positive number M such that M3n is a prime numberMathworldPlanetmath for any integer n>0. This is Mills’ constant. Assuming that the Riemann hypothesisMathworldPlanetmath is true, the constant’s value would be approximately 1.3063778838630806904686144926026 (see A051021 in Sloane’s OEIS). According to Caldwell and Cheng, Mills’ original paper “contained no numerics,” it only proved the existence of such a number; moreover it referred to Mcn, and it is those who hunt for the specific value of M who often choose c=3. Armed with these assignments and assumptionsPlanetmathPlanetmath (including certain probable primesMathworldPlanetmath), Caldwell and Cheng computed almost seven thousand base 10 digits of Mills’ constant. The first few primes generated by Mills’ constant would be 2, 11, 1361, 2521008887, etc. (listed in A051254).


  • 1 C. K. Caldwell & Y. Cheng, “Determining Mills’ Constant and a Note on Honaker’s Problem” J. Integer Sequences 8 (2005): 05.4.1
  • 2 W. H. Mills, “A prime-representing function”, Bull. Amer. Math. Soc. 53 (1947): 604
Title Mills’ constant
Canonical name MillsConstant
Date of creation 2013-03-22 16:42:18
Last modified on 2013-03-22 16:42:18
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A41
Synonym Mills constantMathworldPlanetmath
Synonym Mills’s constant