DOI QR코드

DOI QR Code

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 (Department of Mathematics Education, Mokpo National University)
  • 이헌수 (목포대학교 수학교육과)
  • Received : 2020.09.21
  • Accepted : 2020.11.18
  • Published : 2020.12.31

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.

References

  1. Y.Y.Park and H.S.Lee, "On the several differences between primes," Journal of Applied Mathematics & Computing, Vol.13, No.1-2, pp.37-51, 2003. https://doi.org/10.1007/BF02936073
  2. H.S.Lee and Y.Y.Park, "On the primes with pn+1-pn = 8 and sum of their reciprocals," Journal of Applied Mathematics & Computing, Vol.22, No.1-2, pp.441-452, 2006. https://doi.org/10.1007/BF02896492
  3. H.S.Lee and Y.Y.Park, "The Generaliza tion of Clement's Theorem on Pairs of Primes," Journal of Applied Mathematics & Informatics, Vol.27, No.1-2, pp.89-96, 2009.
  4. Viggo Brun, "La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + …, ou les dénominateurs sont 'nombres premieres jumeaux' est convergente ou finie," Bulletin des sciences mathématiques, Vol.43, pp.100-104, 124-128, 1919.
  5. E.S.Sehmer, "A special summation method in the theory of prime numbers and its application to 'Brun's sum," Nordisk Mat. Tidskr., Vol.24, pp.74-81, 1942.
  6. Froberg. "On the sum of inverses of primes and twin primes," Nordisk Tidskr. Informationsbehandling (BIT), Vol.1, pp. 15-20, 1961.
  7. J.W.Wrench, Jr., "Evaluation of Artin's Constant and the Twin-Prime Constant," Math. Comp., Vol.15, pp.396-398, 1961. https://doi.org/10.1090/S0025-5718-1961-0124305-0
  8. J.Bohman, "Some computational results regarding the prime numbers below 2,000,000,000," Nordisk Tidskr. Informationsbehandling(BIT), Vol.13, p.127, 1974.
  9. R.P.Brent, "The distribution of small gaps between successive primes," Math. Comp., Vol.28, pp.315-324, 1974. https://doi.org/10.1090/S0025-5718-1974-0330017-X
  10. T.Nicely, "Enumeration to 1014 of the Twin primes and Brun's constant," Virginia J. Sci., Vol.46, pp.195-204, 1996.
  11. D.Shin, W.Bae, H.Shin, S.Nam and H.W.Lee, "Crypft+ : Python/PyQt based File Encryption & Decryption System Using AES and HASH Algorithm," Journal of The Korea Internet of Things Society, Vol.2, No.3, pp.43-51, 2016. https://doi.org/10.20465/KIOTS.2016.2.3.043
  12. G.H.Hardy and J.E.Littlewood, "Some problems of "Partitio Numerrorum", III: On the expression of a number as a sum of primes," Acta Math., Vol. 44, pp.1-70, 1923. https://doi.org/10.1007/BF02403921
  13. N.J.A.Sloane, "Sequence A005597 (Decimal expansion of the twin prime constant)," The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2019-11-01.
  14. Robert Joseph Harley, unpublished work, accessed 2005; Web document no longer accessible.
  15. Wolfram Research, The Mathematica Book, 4th ed., Champaign: Wolfram Media, 1993.
  16. H.Riesel, Prime Numbers and Computer Methods for Factorization, 2nd ed., Basel: Birkauser, p.255, 1994.