A prime-representing function

by W. H. MILLS

A function [tex:f(x)] is said to be a prime-representing function if [tex:ƒ(x)]
is a prime number for all positive integral values of x. It will be shown that there exists a real number A such that [tex:A^{3^x}] is a prime-representing function, where [tex:[R\]] denotes the greatest integer less than or equal to R.
Let [tex:p_n] denote the [tex:n]th prime number. 

A. E. Ingham has shown that
(1) [tex:p_{n+1} - p_n \lt Kp_n^{\frac{5}{8}}]
where [tex:K] is a fixed positive integer.

[LEMMA.] If [tex:N] is an integer greater than [tex:K^8] there exists a prime [tex:p] such that [tex: N^3 \lt p \lt (N+1)^3-1].

(PROOF.) Let [tex: p_n] be the greatest prime less than [tex: N^3]. Then
( 2 ) [tex: N^3 \lt p_{n+l} \lt p_n + K p_n^{\frac{5}{8}} \lt N^3 + KN^{\frac{15}{8}} \lt N^3 +N^2 \lt (N + 1)^3 - 1].

Let [tex: P_0] be a prime greater than [tex:K^8]. Then by the lemma we can
construct an infinite sequence of primes, [tex:P_0,P_1, P_2,\cdots], such that
[tex: P_n^2 \lt P_{n+1} \lt (P_n +1)^3 - 1]. Le t
(3) [tex: u_n = P_n^{3^{-n}} , v_n = (P_n + 1)^{3^{-n}}].
Then
(4) [tex: v_n \gt u_n, u_{n+1} = P_{n+1}^{3^{-n-1}} \gt P_n^{3^{-n}} = u_n],
(5) [tex: v_{n+1} = (P_{n+1} + 1)^{3^{-n-1}} \lt (P^n + 1)^{3^{-n}} = v_n].
It follows at once that the [tex:u_n] form a bounded monotone increasing
sequence. Let [tex: A = \lim_{n \to \infty} u_n].

[THEOREM.] [tex:A^{3^x}] is a prime-representing function.
(PROOF.) From (4) and (5) it follows that [tex: u_n \lt A \lt v_n], or [tex: P_n \lt A^{3^x} \lt P_{n+1}] .
Therefore [tex:\[A^{3^x}\] = P_n] and [tex: [A^{3^x}] ] is a prime-representing function. 

https://diavolo666.hatenablog.com/entry/2020/05/09/182552