Search for primes of the form $m^2+1$

TL;DR

This paper reports extensive computational searches for primes of the form m^2+1 up to 10^20, analyzing their distribution, sign changes, and related constants, and formulates conjectures about their properties.

Contribution

It provides the first large-scale computational analysis of primes of the form m^2+1, including sign change behavior and analogs of classical number theory constants.

Findings

Number of sign changes of the difference pi_q(x) - (C_q/2)∫ du/√u log u is investigated.

Analogues of Brun's constant and Skewes number are calculated.

No Chebyshev bias is observed for primes of the form m^2+1.

Abstract

The results of the computer hunt for the primes of the form $q = m^2+1$ up to $10^{20}$ are reported. The number of sign changes of the difference $\pi_q(x) - \frac{C_q}{2}\int_2^x{du \over \sqrt{u}\log(u)}$ and the error term for this difference is investigated. The analogs of the Brun's constant and the Skewes number are calculated. An analog of the B conjecture of Hardy--Littlewood is formulated. It is argued that there is no Chebyshev bias for primes of the form $q=m^2+1$. All encountered integrals we were able to express by the logarithmic integral.