Math’s Bite: Mills’ Constant

In number theory, Mills’ theorem states that there exists a real constant A such that |_A^(3^n)_| is prime for all positive integers n (note that this is a floor function). Mills’ constant is defined as the smallest real positive number such that Mills’ theorem is true.

This constant is named after William H. Mills, who proved the existence of A based on results of Guido Hoheisel and Albert Ingham on the prime gaps in 1947.

If Riemann’s hypothesis is true, Mills’ constant is approximately:

 theta=1.306377883863080690...

Mills’ Primes

The primes generated using Mills’ constant are known as Mills’ primes:

“If ai denotes the ith prime in this sequence, then ai can be calculated as the smallest prime number larger than a_{i-1}^3. In order to ensure that rounding A^{3^n} produces this sequence of primes, it must be the case that a_i < (a_{i-1}+1)^3.”

In 2005, Caldwell and Cheng computed 6850 base 10 digits of Mills’ constant under the assumption that the Riemann hypothesis is true.

Hope you enjoyed this installment of ‘Math’s Bite’! M x

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