An Equivalent Form of Strong Lemoine Conjecture and Several Relevant Results

  • Shaohua ZhangEmail author


In this paper, we consider some problems involving Strong Lemoine Conjecture in additive number theory. Based on Dusart’s inequality and Rosser-Schoenfeld’s inequality, we obtain several new results and give an equivalent form of Strong Lemoine Conjecture.

Key words

Lemoine Conjecture Dusart’s inequality Rosser-Schoenfeld’s inequality Euler totient function primecounting function 

CLC number

O 156 


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The author is particularly indebted to the referees for their comments improving the proof of the Theorem 2. Thank LIU Shuhua for verifying that every odd n can be represented as 2x+y for 105<n<3×109, where x and y are coprime composite numbers, and for checking Strong Lemoine Conjecture on a computer to be valid for all odd n up to 109.


  1. [1]
    Dickson L E. History of the Theory of Numbers[M]. New York: Chelsea, 1952.Google Scholar
  2. [2]
    Hardy G H, Littlewood J E. On some problems of “partitio numerorum” III: On the expression of a number as a sum of primes [J]. Acta Math, 1923, (44): 1–70.CrossRefGoogle Scholar
  3. [3]
    Chen J R. On the representation of a larger even integer as the sum of a prime and the product of at most two primes [J]. Sci Sinica, 1973, (16): 157–176.Google Scholar
  4. [4]
    Tao T. Every odd number greater than 1 is the sum of at most five primes[J]. Math Comp, 2014, 83(286): 997–1038.CrossRefGoogle Scholar
  5. [5]
    Helfgott H. The ternary Goldbach conjecture [J]. Gac R Soc Mat Esp, 2013, 16(4): 709–726.Google Scholar
  6. [6]
    Zhao L L. On ternary problems in additive prime number theory [J]. J Number Theory, 2017, (178): 179–189.CrossRefGoogle Scholar
  7. [7]
    Banks W D. Zeta functions and asymptotic additive bases with some unusual sets of primes [J]. Ramanujan J, 2018, 45(1): 57–71.CrossRefGoogle Scholar
  8. [8]
    Kiltinen J O, Young P B. Goldbach, Lemoine, and a know/don't know problem [J]. Math Mag, 1985, 58(4): 195–203.CrossRefGoogle Scholar
  9. [9]
    Sun Z W. On sums of primes and triangular numbers [EB/OL]. [2018-04-17].
  10. [10]
    Dusart P. Estimates of some functions over primes without R.H. [EB/OL]. [2018-04-17]. 0442.
  11. [11]
    Rosser J B, Schoenfeld L. Approximate formulas for some functions of prime numbers [J]. Illinois J Math, 1962, (1): 64–94.CrossRefGoogle Scholar
  12. [12]
    Tatuzawa T. On Bertrand's problem in an arithmetic progression [J]. Proc Japan Acad, 1962, (38): 293–294.CrossRefGoogle Scholar

Copyright information

© Wuhan University and Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsYangtze Normal UniversityChongqingChina

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