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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) In the late forties Mills proved [Mills47] that: Mills' Theorem: there is a real number A for which [] is always a prime (n = 1,2,3,...). The key to finding a value for A is that we need to construct a sequence of primes (Mills' primes) p_{1}, p_{2}, ...; for which p_{i1} is between p_{i}^{3} and (p_{i}+1)^{3}. The constant A is then the limit of p_{n}^{3n} as n approaches infinity. Mills' one page article contained no numerics, it only proved the existence of A. Others [Wright1954] later proved that there are uncountably many choices for A, but again gave no value for A. In fact, we can only prove there are primes between consecutive cubes if we are looking at numbers beyond 10^{6000000000000000000} [Cheng2003a]. If we assume the Riemann Hypothesis, then it is easy to show there are primes between consecutive cubes of integers [CC2005] greater than one; and then we can calculate example to our hearts content. Following tradition (and definitley not following Mills) we begin our sequence of primes as follows: b_{1} = 2,These are called the Mills Primes. This sequence is formed by choosing the minimal prime at each step, and yields the smallest possible value for Mills' constant: 1.30637788386308069046861449260260571291678...A few more Mills primes are known. To make them easier to present, let b_{n+1} = b_{n}^{3}+a_{n}. The sequence a_{n} begins: 3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220 (b_{n} prp), 66768 (b_{n} prp)The primality of b_{7}, b_{8}, and b_{9} (2285 digits) were proved byBouk de Water in 2000; and Morain proved b_{9} (6854 digits) prime in 2005. Carmody found the two PRP's (20562 and 61684 digits) in 2004.
See Also: MillsTheorem, MillsConstant Related pages (outside of this work) References:
Chris K. Caldwell © 19992019 (all rights reserved)
