W. H. MILLS
A function is said to be a prime-representing function if is a prime number for all positive integral values of . It will be shown that there exists a real number such that is a prime-representing function, where denotes the greatest integer less than or equal to .
Let denote the th prime number.
A. E. Ingham has shown that
where is a fixed positive integer.
Lemma
If is an integer greater than there exists a prime such that .
Let be the greatest prime less than . Then
.
Let be a prime greater than . Then by the lemma we can construct an infinite sequence of primes, , such that
.
Let
3..
Then
4.,
5..
It follows at once that the form a bounded monotone increasing sequence. Let .
is a prime-representing function.
From (4) and (5) it follows that , or .
Therefore and is a prime-representing function.