# Mills’ constant

Find the smallest real positive number $M$ such that $\lfloor M^{3^{n}}\rfloor$ is a prime number for any integer $n>0$. This is Mills’ constant. Assuming that the Riemann hypothesis 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 $\lfloor M^{c^{n}}\rfloor$, and it is those who hunt for the specific value of $M$ who often choose $c=3$. Armed with these assignments and assumptions (including certain probable primes), 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).

## References

• 1 C. K. Caldwell & Y. Cheng, “Determining Mills’ Constant and a Note on Honaker’s Problem” 8 (2005): 05.4.1
• 2 W. H. Mills, “A prime-representing function”, Bull. Amer. Math. Soc. 53 (1947): 604
Title Mills’ constant MillsConstant 2013-03-22 16:42:18 2013-03-22 16:42:18 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A41 Mills constant Mills’s constant