Mills' constant

In the late forties Mills [Mills47] proved that there was a real number A>1 for which [] is always a prime (n = 1,2,3,...).  He proved existence only, and did not attempt to find such an A.  Later others [Wright1954] proved that there are uncountably many choices for A, but again gave no value for A.

It is still not yet possible to calculate a proven value for A, but if you are willing to accept the Riemann Hypothesis, then the least possible value for Mills constant (usually called "the Mills Constant") begins as follows:

         1.3063778838 6308069046 8614492602 6057129167 8458515671 3644368053 7599664340 5376682659 8821501403 7011973957 0729696093 8103086882 2388614478 1635348688 7133922146 1943534578 7110033188 1405093575 3558319326 4801721383 2361522359 0622186016 1085667905 7215197976 0951619929 5279707992 5631721527 8412371307 6584911245 6317518426 3310565215 3513186684 1550790793 7238592335 2208421842 0405320517 6890260257 9344300869 5290636205 6989687262 1227499787 6664385157 6619143877 2844982077 5905648255 6091500412 3788524793 6260880466 8815406437 4425340131 0736114409 4137650364 3793012676 7211713103 0265228386 6154666880 4874760951 4410790754 0698417260 3473107746 7757406400 7810935083 4214374426 5420408531

See Also: MillsTheorem, MillsPrime

References:

CC2005

C. Caldwell and Y. Cheng, "Determining Mills' constant and a note on Honaker's problem," J. Integer Seq., 8:4 (2005) Article 05.4.1, 9 pp. (electronic).  Available from http://www.cs.uwaterloo.ca/journals/JIS/.  MR2165330 (Abstract available)

Mills47

W. H. Mills, "A prime-representing function," Bull. Amer. Math. Soc., 53 (1947) 604.  MR 8,567d (Annotation available)

Wright1954

Wright, E. M., "A class of representing functions," J. London Math. Soc., 29 (1954) 63--71.  MR 15,288d