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.
Categories: math.NT