Wright's Fourth Prime

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.

Ancillary files contain primality certificates for two 4932-digit primes. P4932Proof.txt has a primality certificate from primo. This is a text file with PC-style end of line characters. cert2To16382minus35411.txt has a primality certificate from PARI/GP. This file is one (long) line of text