THE THOUGHT ON THE “TWINS”PROBLEM
Journal Title: Дидактика математики: проблемы и исследования - Year 2018, Vol 1, Issue 47
Abstract
We say that the natural numbers p_1 and p_2 are prime twins if they both are prime and p_2-p_1=2. In this paper we consider and discuss some approaches to the well-known problem about infinity of the set of prime twins. We study the function B(N) which is equal to the number of all pairs (p_1,p_2) of prime twins such that p_2≤N. For investigation of behavior of B(N) as N→∞ we use the circular method by Hardy and Littlwood. In accordance of this method, we express B(N) as a trigonometric integral and introduce the sets of “big arcs” and “small arcs”, respectively. The integration over the set of big arcs leads to the term of the form CN/〖ln〗^2N +o(N/〖ln〗^2N )as N→∞. Therefore, the main problem is to estimate the integral over the set of small arcs. We consider various methods of the estimate and show that some of them are highly nontrivial.
Authors and Affiliations
Valery Volchkov, Vitaly Volchkov, Pavlo Masharov
Keywords
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