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Conjectures on Representations Involving Primes

  • Zhi-Wei Sun
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 220)

Abstract

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer n, there exists \(k\in \{0,\ldots ,n\}\) such that \(n+k\) and \(n+k^2\) are both prime. (ii) Each integer \(n>1\) can be written as \(x+y\) with \(x,y\in \{1,2,3,\ldots \}\) such that \(x+ny\) and \(x^2+ny^2\) are both prime. (iii) For any rational number \(r>0\), there are distinct primes \(q_1,\ldots ,q_k\) with \(r=\sum _{j=1}^k1/(q_j-1)\). (iv) Every \(n=4,5,\ldots \) can be written as \(p+q\), where p is a prime with \(p-1\) and \(p+1\) both practical, and q is either prime or practical. (v) Any positive rational number can be written as m/n, where m and n are positive integers with \(p_m+p_n\) a square (or \(\pi (m)\pi (n)\) a positive square), \(p_k\) is the kth prime and \(\pi (x)\) is the prime-counting function.

Keywords

Conjectures Primes Practical numbers Representations 

2010 Mathematics Subject Classification.

Primary 11A41 11P32 Secondary 11B13 11D68 

Notes

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant No. 11571162).

References

  1. 1.
    J.R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16, 157–176 (1973)MathSciNetzbMATHGoogle Scholar
  2. 2.
    R. Crandall, C. Pomerance, Prime Numbers: A Computational Perspective, 2nd edn. (Springer, New York, 2005)zbMATHGoogle Scholar
  3. 3.
    R. Crocker, On the sum of a prime and two powers of two. Pac. J. Math. 36, 103–107 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    P. Erdős, I. Niven, Some properties of partial sums of the harmonic series. Bull. Am. Math. Soc. 52(4), 248–251 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    O. Gerard, Z.-W. Sun, Refining Goldbach’s conjecture by using quadratic residues. A Message to Number Theory List (November 19, 2012), http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;c08d598.1211
  6. 6.
    R.K. Guy, Unsolved Problems in Number Theory, 3rd edn. (Springer, New York, 2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    B. Green, T. Tao, The primes contain arbitrary long arithmetic progressions. Ann. Math. 167, 481–547 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D.R. Heath-Brown, Primes represented by \(x^3+2y^3\). Acta Math. 186, 1–84 (2001)Google Scholar
  9. 9.
    H.A. Helfgott, The ternary Goldbach problem (2015), arXiv:1501.05438
  10. 10.
    E. Lemoine, L’intermédiare des mathématiciens 1, 179 (1894); ibid 3, 151 (1896)Google Scholar
  11. 11.
    M. Margenstern, Les nombres pratiques: théorie, observations et conjectures. J. Number Theory 37, 1–36 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    G. Melfi, On two conjectures about practical numbers. J. Number Theory 56, 205–210 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Murthy, Sequence A109909 in OEIS (On-Line Encyclopedia of Integer Sequences) and comments for A034693 in OEIS, http://www.oeis.org
  14. 14.
    M.B. Nathanson, in Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics, vol. 164 (Springer, 1996)Google Scholar
  15. 15.
    B.M. Stewart, Sums of distinct divisors. Am. J. Math. 76, 779–785 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Z.-W. Sun, Sequences A185636, A199920, A204065, A208244, A209236, A209253, A209312, A209315, A209320, A210479, A210480, A210681, A218654, A218656, A218754, A218825, A218829, A219864, A220272, A220413, A220431, A220554, A220572, A224030, A227898, A227899, A227909 (joint with O. Gerard), A227923, A228425, A229166, A230217, A230219, A230223, A230224, A230230, A230241, A230243, A230252, A230254, A230351, A230494, A230507, A230516, A231201, A231516, A231555, A231557, A231561, A231631, A231633, A231635, A231725, A231883, A232109, A232174, A232186, A232269, A232353, A233386, A232398, A233544, A233547, A233654, A233793, A233867, A233918, A234200, A234246, A234308, A234309, A234347, A234359, A234360, A234451, A234504, A234808, A234809, A235189, A235592, A235613, A235614, A235661, A235703, A235987, A236358, A236567, A236968, A236998, A237049, A237050, A237123, A237127, A237130, A237168, A237183, A237184, A237253, A237523, A237524, A237715, A238386, A238405, A238703, A238732, A238733, A241844, A242425, A242441, A255677, A256555, A256707, A257364, A257856, A257922, A257924, A257926, A257928, A257938, A258803, A258836, A258838, A259487, A259488, A259492, A259531, A259539, A259540, A259628, A259678, A259712, A259789, A259915, A259916, A260078, A260080, A260082, A260120, A260140, A260232, A260252, A260753, A260886, A260888, A261136, A261281, A261295, A261339, A261354, A261352, A261353, A261361, A261362, A261382, A261385, A261387, A261395, A261437, A261462, A261513, A261515, A261528, A261533, A261541, A261583, A261627, A261628, A261641, A261653, A262311, A262785, A262982, A262985, A263992, A263998, A264010, A264025 in OEIS, http://www.oeis.org
  17. 17.
    Z.-W. Sun, Mixed sums of squares and triangular numbers. Acta Arith. 127, 103–113 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Z.-W. Sun, On sums of primes and triangular numbers. J. Comb. Number Theory 1, 65–76 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Z.-W. Sun, Conjectures involving arithmetical sequences. In: Number Theory: Arithmetic in Shangri-La, ed. by S. Kanemitsu, H. Li, J. Liu, Proceedings of 6th China-Japan Seminar, Shanghai, August 15–17, 2011 (World Sci., Singapore, 2013), pp. 244–258Google Scholar
  20. 20.
    Z.-W. Sun, On functions taking only prime values. J. Number Theory 133, 2794–2812 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Z.-W. Sun, On \(a^n + bn\) modulo \(m\), arXiv:1312.1166
  22. 22.
    Z.-W. Sun, Problems on combinatorial properties of primes. In: Plowing and Starring through High Wave Forms, ed. by M. Kaneko, S. Kanemitsu, J. Liu, Proceedings of 7th China-Japan Seminar on Number Theory, Fukuoka,  Oct. 28–Nov. 1, 2013, Ser. Number Theory Applications, vol. 11 (World Sci., Singapore, 2015), pp. 169–187Google Scholar
  23. 23.
    Z.-W. Sun, A new theorem on the prime-counting function. Ramanujan J. 42, 59–67 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    I.M. Vinogradov, The Method of Trigonometrical Sums in the Theory of Numbers (Dover, New York, 2004)zbMATHGoogle Scholar
  25. 25.
    A. Weingartner, Practical numbers and the distribution of divisors. Q. J. Math. 66, 743–758 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Y. Zhang, Bounded gaps between primes. Ann. Math. 179, 1121–1174 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

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