FANDOM


The lowest value of Mills' constant assuming the Riemann Hypothesis is true. In 1947 W. H. Mills proved that there is a constant K such that, for all values of n, the integer part of K3n is prime. The first 100 digits of the constant are: 1.3063778838 6308069046 8614492602 6057129167 8458515671 3644368053 7599664340 5376682659 8821501403 7011973957...

Here is what you get for the first few values of n:

K3 = 2.229494... 
K32 = 11.082031... 
K33 = 1361.000001... 
K34 = 2521008887.000000... 
K35 = 16022236204009818131831320183.000000...

Each of these numbers is the cube of the previous one, and when the fraction is removed the resulting integer is prime. The sequence: 2, 23+3=11, (23+3)3+30=1361, ... is Sloane's A051254.

It is pretty easy to see that Mills' theorem seems to be true, simply because there are so many primes. For example, start with 3: 33 is 27. There are several primes between 27 and 43=64, of which the first is 29. 293 is 24389 and 303 is 27000 — there are even more primes to choose from this time. Choosing the first available prime each time, we get the sequence 3, 29, 24391, 14510715208481, 3055388613462301256452407743005777548691, .... The constant K in this case would be approximately 3055388613462301256452407743005777548691(1/243)= 1.45375086254.... In a similar manner, starting with 5 we get 127, 2048413, 8595132382702380079, 634976584256084664026852011723442922433087739799461233111, ... No matter what prime one starts with, there are plenty of primes to choose from each time and therefore plenty of possible values for Mills' constant.

The difficulty in proving this for certain comes from the fact that it is difficult to prove that there is a prime between any two consecutive cubes. So far that has only been proven for primes up to 106000000000000000000.

Community content is available under CC-BY-SA unless otherwise noted.