Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let be the th prime, then there exists
a constant
such that

(2)

for all .
This has more recently been strengthened to

(3)

(Mozzochi 1986). If the Riemann hypothesis
is true, then Cramér (1937) showed that

(4)

(Finch 2003).

Hardy and Wright (1979) and Ribenboim (1996) point out that, despite the beauty of such prime formulas, they do not have any practical
consequences. In fact, unless the exact value of is known, the primes themselves
must be known in advance to determine .

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