Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture
Jon Grantham, Hester Graves
TL;DR
This work computationally enumerates primes of the form $p=m^2+1$ with $p<6.25\times 10^{28}$ via a triple-sieve method and uses the resulting data to verify Goldbach's 'other other' conjecture up to the bound. It introduces the notion of Goldbach champions through the sequence $A= \{a: a^2+1\text{ prime}\}$ and the metric $j(a_n)$, and derives conditional results on the growth of $j(n)$ under Schinzel's Hypothesis H and the Bateman–Horn Conjecture. The paper proves, under Hypothesis H, that $j(a_n)>1$ occurs infinitely often, while BH implies $\limsup_{n\to\infty} j(a_n)=\infty$, and shows there are infinitely many $a_n$ with both $a_n^2+1$ and $(a_n-2)^2+1$ prime, forcing a regular occurrence of small gaps in $A$. These results advance understanding of the distribution of primes in $m^2+1$ and their additive representations, with broader implications for conjectures in prime values of polynomials and cyclotomic generalizations.
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.