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

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

2008-03-10

arxiv.org/abs/0803.1456v3

Mathematics

Number Theory

The Fig.10 is added. The n=1 is not included into the definition of the logarithmic density

Scientific paper

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.

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