Primes between 8th powers?

Abstract

OpenAI (using a non-public "internal frontier model'') has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH). While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true) their implications would be. While there is widespread agreement that any version of the QRH will provide various improved bounds on the location of the prime numbers, I feel that there is some value in making these improvements explicit. Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\in \mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$. All of this is contingent on the validity of the claimed 7/8 version of the QRH.