Wright's Fourth Prime

TL;DR

This paper explores specific initial values in Wright's sequence that generate prime numbers at each step, including a new initial value that produces a fourth prime, and discusses the transcendental nature of the sum of their reciprocals.

Contribution

It identifies a precise initial value that yields a prime at the fourth step in Wright's sequence, extending previous results and analyzing the sequence's properties.

Findings

A specific initial value c produces four primes in Wright's sequence.

The fourth prime in the sequence is a large 4932-digit prime.

The sum of reciprocals of the primes in Wright's sequence is transcendental.

Abstract

Wright proved that there exists a number $c$ such that if $g_0 = c$ and $g_{n+1} = 2^{g_n}$, then $\lfloor g_n \rfloor$ is prime for all $n > 0$. Wright gave $c = 1.9287800$ as an example. This value of $c$ produces three primes, $\lfloor g_1 \rfloor = 3$, $\lfloor g_2 \rfloor = 13$, and $\lfloor g_3 \rfloor = 16381$. But with this $c$, $\lfloor g_4 \rfloor$ is a 4932-digit composite number. However, this slightly larger value of $c$, \[ c = 1.9287800 + 8.2843 \cdot 10^{-4933}, \] reproduces Wright's first three primes and generates a fourth: \[ \lfloor g_4 \rfloor = 191396642046311049840383730258 \text{ } \ldots \text{ } 303277517800273822015417418499 \] is a 4932-digit prime. Moreover, the sum of the reciprocals of the primes in Wright's sequence is transcendental.