  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.
