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

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.