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A study on the approximation function for pairs of primes with difference 10 between consecutive primes

연속하는 두 소수의 차가 10인 소수 쌍에 대한 근사 함수에 대한 연구

Lee, Heon-Soo
이헌수


Abstract

In this paper, I provided an approximation function Li*2,10(x) using logarithm integral for the counting function π*2,10(x) of consecutive deca primes. Several personal computers and Mathematica were used to validate the approximation function Li*2,10(x). I found the real value of π*2,10(x) and approximate value of Li*2,10(x) for various x≤1011. By the result of theses calculations, most of the error rates are margins of error of 0.005%. Also, I proved that the sum C2,10(∞) of reciprocals of all primes with difference 10 between primes is finite. To find C2,10(∞), I computed the sum C2,10(x) of reciprocals of all consecutive deca primes for various x≤1011 and I estimate that C2,10(∞) probably lies in the range C2,10(∞)=0.4176±2.1×10-3.