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arXiv · 0803.1456

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

Marek Wolf
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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.

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Marek Wolf. 2008-03-10. Search for primes of the form $m^2+1$. https://arxiv.org/abs/0803.1456

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