Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture

TL;DR

This paper computationally verifies a lesser-known Goldbach-related conjecture by listing primes of the form m^2+1 up to 6.25×10^28 and introduces 'Goldbach champions' with conditional proofs based on major conjectures.

Contribution

It provides extensive computational data on primes of the form m^2+1 and introduces the concept of 'Goldbach champions' with conditional theoretical results.

Findings

Verified the conjecture up to 6.25×10^28

Introduced 'Goldbach champions' as a new concept

Proved conditional results assuming major hypotheses

Abstract

We compute all primes up to $6.25\times 10^{28}$ of the form $m^2+1$. Calculations using this list verify, up to our bound, a less famous conjecture of Goldbach. We introduce `Goldbach champions' as part of the verification process and prove conditional results about them, assuming either Schinzel's Hypothesis H or the Bateman-Horn Conjecture.