A prime-representing function - Diavolo666’s diary

by W. H. MILLS

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

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

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

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

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

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