Initial day: July 31, 2001      Last modified: July 27, 2017
    visits since April 10, 2002            

      Zhi-Wei Sun
      Department of Mathematics
      Nanjing University
      Nanjing 210093
      People's Republic of China
      Office in the Dept.: Room 103

              Last Name Sun
              First Name Zhi Wei
              Year of Birth 1965

Research Interests

    Number Theory (especially Combinatorial Number Theory),
    Combinatorics, Group Theory, Mathematical Logic.

Academic Service

    Editor-in-Chief of Journal of Combinatorics and Number Theory, 2009--.
     You may submit your paper by sending the pdf file to
     or to one of the two managing editors Florian Luca and Jiang Zeng. (A sample tex file)

    Reviewer for Zentralblatt Math., 2007--.
    Reviewer for Mathematical Reviews, 1992--.
    Member of the American Mathematical Society, 1993--.
    Referee for Adv. in Math., Proc. Amer. Math. Soc., Acta Arith., J. Number Theory, J. Combin. Theory Ser. A, European J. Combin.,
     Finite Fields Appl., Adv. in Appl. Math., Discrete Math., Discrete Appl. Math., Ramanujan J., SIAM Review
    I refuse to referee for any open access journal which asks for page charges.

School Education and Employment History

   1980.9--1983.7 The High Middle School Attached to Nanjing Normal Univ.
   1983.9--1992.6 Department of Mathematics, Nanjing University
             (Undergraduate--Ph. D. Candidate; B. Sc. 1987, Ph. D. 1992)
   1992.7--     Teacher in Department of Mathematics, Nanjing University
   1994.4--1998.3 Associate Professor in Math.
   1998.4--     Full Professor in Math.
   1999.11-     Supervisor of Ph. D. students

My paper Further Results on Hilbert's Tenth Problem (based on my PhD thesis in 1992)

My 24-Conjecture with $2400 Prize (see also OEIS A281976 and arXiv:1701.05868

   Any natural number n can be written as the sum of squares of four nonnegative integers x, y, z and w such that both x and x+24y are squares. [This has been verified for n up to 1010 by Qing-Hu Hou.]

My 1-3-5 Conjecture with $1350 Prize (see also OEIS A271518 and arXiv:1604.06723

   Any natural number n can be written as the sum of squares of four nonnegative integers x, y, z and w such that x+3y+5z is also a square. [This has been verified for n up to 1010 by Qing-Hu Hou.]

My Conjecture involving Primes and Powers of 2 with $1000 Prize (see also Conjecture 3.6(i) of this paper)

   Every n = 2, 3, ... can be written as a sum of two positive integers k and m such that 2k + m is prime. [This has been verified for n up to 107.]

My Conjecture on Unit Fractions involving Primes with $500 Prize (see also Conjecture 4.1(i)-(ii) of this paper)

   Let d be -1 or 1. Each positive rational number can be written as 1/(p(1)+d) + 1/(p(2)+d) + ... + 1/(p(k)+d), where p(1),...,p(k) are distinct primes.

My Conjecture on Primes of the Form x2+ny2 with $200 Prize (see also Conjecture 2.21(i) of this paper)

   Each n = 2,3,... can be written as x+y with x and y positive integers such that x+ny and x2+ny2 are both prime.

My Conjecture related to Bertrand's Postulate with $100 Prize (see also Conjecture 2.18 of this paper)

   Let n be any positive integer. Then, for some k=0,...n, both n+k and n+k2 are prime. [I have verified this conjecture for n up to 200,000,000.]

My 100 Conjectures on Representations involving Primes or related Things

My Little 1-3-5 Conjecture with $135 Prize (see this paper for more such conjectures)

   Each n = 0,1,2,... can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z nonnegative integers. [I have proved the weaker version with x,y,z integers.]

My 60 Open Problems on Combinatorial Properties of Primes

My Conjecture on the Prime-Counting Function (see Conjectures 2.1, 2.6 and 2.22 of this paper)

 (i) For any integer n>1, π(k*n) is prime for some k = 1,...,n, where π(x) denotes the number of primes not exceeding x. [I have verified this for n up to 107. See OEIS A237578.]
 (ii) For every positive integer n, π(π(k*n)) is a square for some k = 1,...,n. [I have verified this for n up to 2*105. See OEIS A238902 and OEIS A239884.]
 (iii) For each integer n>2, π(n-p) is a square for some prime p < n. [I have verified this for n up to 5*108. See OEIS A237706 and OEIS A237710.]

My "Super Twin Prime Conjecture" (see Conjecture 3.2 of this paper)

   Each n = 3, 4, ... can be written as k + m with k and m positive integers such that p(k) + 2 and p(p(m)) + 2 are both prime, where p(j) denotes the j-th prime. [I have verified this for n up to 109.]

My Conjecture on Unification of Goldbach's Conjecture and the Twin Prime Conjecture (see Conjecture 3.1 of this paper)

   Any even number greater than 4 can be written as p + q with p, q and prime(p+2) + 2 all prime, where prime(n) denotes the n-th prime.

My Conjecture involving the n-th Prime (see also Conjecture 4.4 of this paper)

   For each m = 1,2,3, ... there is a positive integer n such that m + n divides pm + pn, where pk denotes the k-th prime. Moreover, we may require that n < m*(m-1) if m > 2. [I have verified this for m up to 105, and also formulated many other similar conjectures.]

My Conjecture on Recurrence for Primes (see also Conj. 1.2 of this published paper)

   For any positive integer n different from 1,2,4,9, the (n+1)-th prime pn+1 is just the least positive integer m such that 2sk2 (k=1,...,n) are pairwise distinct modulo m, where sk = pk-pk-1+...+(-1)k-1p1. [I have verified this for n=1,...,100000.]

My Conjecture on Alternating Sums of Consecutive Primes (see also Conj. 1.3 of this published paper)

   For any positive integer m, there are consecutive primes pk,...,pn (k < n) not exceeding 2m+2.2*sqrt(m) such that m = pn-pn-1+...+(-1)n-kpk. [I have verified this for m up to 105.]

My Conjecture on Fibonacci Quadratic Nonresidues (see OEIS A241568, OEIS A241604 and A241675)

   For each n = 5,6,..., there exists a Fibonacci number F(k) < n/2 such that no square is congruent to F(k) modulo n. [This can be easily reduced to the case with n prime. I have verified it for all primes p with 3 < p < 3*109.]

My Conjecture on Primitive Roots of Special Forms (see OEIS A239957, A241476, A241504 and A241516)

   Any prime p has a primitive root g < p modulo p with g-1 a square. Also, each prime p has a primitive root g < p modulo p which is a partition number. [I verified the two assertitions for all primes below 107 and 2*107 respectively. Later, C. Greathouse extended the verification of the first assertion in the conjecture to all primes below 1010.]

My Conjecture on Representations involving Cubes (see also Conjecture 1.2(iii) of this paper)

   Any positive integer n can be written as the sum of a nonnegative cube, a square and a positive triangular number. [I have verified this for n up to 2*107.]

My Conjecture on Representations involving Fourth Powers (see also Conjecture 1.2(iv) of this paper)

   Any nonnegative integer n can be written as x4 + y(3y+1)/2 + z(7z+1)/2 with x,y,z integers.

My Problem Upgrading Waring's Problem (see OEIS A271099, OEIS A271169, A271237, A267826 and A267861)

My 18 Conjectures in Additive Combinatorics

   Let A be a subset of an additive abelian group G with |A|=n>3. Then there is a numbering a1, ..., an of all the elements of A such that a1+a2+a3, ..., an-1+an+a1, an+a1+a2 are pairwise distinct. [We have proved this for any torsion-free abelian group G, see also A228772 in OEIS.]

My Conjectures on Determinants (see also arXiv:1308.2900 and Three mysterious conjectures on Hankel-type determinants)

My 100 Open Conjectures on Congruences

My 234 Conjectural Series for Powers of π and Other Constants (Announcements: 1, 2, 3, 4, 5, 6)

   Let C(n,k) denote the binomial coefficient n!/(k!(n-k)!) and let Tn(b,c) denote the coefficient of xn in (x2+bx+c)n. Then
     k ≥ 0 (126k+31)Tk(22,212)3/(-80)3k = 880*sqrt(5)/(21π),
     k ≥ 0 (24k+5)C(2k,k)Tk(4,9)2/282k = 49(sqrt(3)+sqrt(6))/(9π),
     k ≥ 0(2800512k+435257)C(2k,k)Tk(73,576)2 /4342k = 10406669/(2sqrt(6)π),
     k>0(28k2-18k+3)(-64)k /(k5C(2k,k)4C(3k,k)) = -14n>01/n3.
     n ≥ 0(28n+5)24-2n C(2n,n)k ≥ 0 5k C(2k,k)2C(2(n-k),n-k)2/C(n,k) = 9(sqrt(2)+2)/π.
     n ≥ 0 (18n2+7n+1)(-128)-n C(2n,n)2k ≥ 0 C(-1/4,k)2C(-3/4,n-k)2 = 4*sqrt(2)/π2.
     n ≥ 0(40n2+26n+5)(-256)-n C(2n,n)2k ≥ 0 C(n,k)2C(2k,k)C(2(n-k),n-k) = 24/π2.

My Hypothesis on the Parities of Ω(n)-n (see also a public message and arXiv:1204.6689)

   We have |{n ≤ x: n-Ω(n) is even}| > |{n ≤ x: n-Ω(n) is odd}| for any x ≥ 5, where Ω(n) denotes the total number of prime factors of n (counted with multiplicity). Moreover, n ≤ x(-1)n-Ω(n) > sqrt(x) for any x ≥ 325. [I have shown that the hypothesis implies the Riemann Hypothesis, and verified it for x up to 1011.]

My Conjecture on Sums of Primes and Triangular Numbers

   Each natural number not equal to 216 can be written in the form p+Tx , where p is 0 or a prime, and Tx=x(x+1)/2 is a triangular number. [This has been verified up to 1,000,000,000,000.] In general, for any a,b=0,1,2,... and odd integer r, all sufficiently large integers can be written in the form 2ap +Tx , where p is either zero or a prime congruent to r modulo 2b.

My Conjecture on Sums of Polygonal Numbers

   For each integer m>2, any natural number n can be expressed as pm+1(x1) + pm+2(x2) + pm+3(x3) + r with x1,x2,x3 nonnegative integers and r among 0,...,m-3, where pk(x)=(k-2)x(x-1)/2+x (x=0,1,2,...) are k-gonal numbers. In particular, every natural number is the sum of a square, a pentagonal number and a hexagonal number. [For m=3, m=4,...,10, and m=11,...,40, this has been verified for n up to 30,000,000, 500,000 and 100,000 respectively.]

My Conjecture on Disjoint Cosets (see Conjecture 1.2 of this published paper)

   Let a1G1 , ..., akGk (k>1) be finitely many pairwise disjoint left cosets in a group G with all the indices [G:Gi] finite. Then, for some distinct i and j the greatest common divisor of [G:Gi] and [G:Gj] is at least k.

My Conjecture on Covers of Groups

   Let a1G1 , ..., akGk be finitely many left cosets in a group G which cover all the elements of G at least m>0 times with ajGj irredundant. Then k is at least m+f([G:Gj]), where f(1)=0 and f(p1 ... pr) =(p1-1) + ... +(pr-1) for any primes p1 , ..., pr .

My Conjecture on Linear Extension of the Erdos-Heilbronn Conjecture

Redmond-Sun Conjecture (in PlanetMath.)

   Papers Indexed in SCI or SCI-E Papers Listed by Field
   Recent Publications (2008-) Preprints on arXiv
   Publications during 2000-2007 Publications during 1987-1999

Other Information
   Research Grants Awards and Honours
   Academic Visits Courses Taught and Ph.D Students
   Notes on Some Conjectures of Z. W. Sun Introduction to Sun's Papers on Covers
   Books and Papers Citing Sun's Work Webpages of Wall-Sun-Sun Prime [1, 2, 3]
   Covers, Sumsets and Zero-sums Link to the useful Number Theory Web
   Mixed Sums of Primes and Other Terms Articles on arXiv: Combinatorics, Number Theory

Invited Lectures in Mathematics
  1. Simple Ideas for Famous Problems, 1996.
  2. Various Number-theoretic Quotients and Related Congruences, 2000.
  3. Recent Progress on Covers of the Integers and Their Applications, 2000.
  4. On Hilbert's Tenth Problem and Related Topics, 2000.
  5. Recent Progress on Combinatorial Number Theory (Chinese), 2001.
  6. Equalities and Inequalities Related to Covers of Z or Groups, 2002.
  7. On the Structure of Periodic Arithmetical Maps, 2002.
  8. New Results on Subset Sums, 2002.
  9. On Zero-sum Problems, 2002.
  10. On the Sum $\sum_{k\equiv r (mod m)}\binom nk$ and Related Results, 2002.
  11. Introduction to Bernoulli and Euler Polynomials, 2002.
  12. Problems and Results in Combinatorial Number Theory, 2002.
  13. Sumsets with Polynomial Restrictions, 2002.
  14. The Magic of Mathematics (Chinese), 2002, 2006.
  15. How to Unify Covering Systems, Restricted Sumsets and Zero-sum Problems, 2003.
  16. Recent Progress on Zero-sum Problems and Snevily's Conjecture, 2004.
  17. Covering Systems and their Connections to Zero-sums, 2004.
  18. On Disjoint Systems of Residue Classes or Cosets of Subgroups, 2004.
  19. On Various Combinatorial Sums and Related Identities, 2004.
  20. Two Local-Global Theorems and a Powerful Formula, 2004.
  21. Groups and Combinatorial Number Theory, 2004.
  22. On Some Conjectures of Erdos-Heilbronn, Lev and Snevily, 2004, 2005.
  23. Problems and Results on Covering Systems, 2005.
  24. Some Congruences Motivated by Algebraic Topology, 2005.
  25. Some Curious Results on Bernoulli and Euler Polynomials, 2005, 2006.
  26. A Survey of Zero-sum Problems on Abelian Groups, 2006.
  27. Recent Progress on Congruences involving Binomial Coefficients, 2006.
  28. Covering Systems and Periodic Arithmetical Functions, 2006.
  29. Combinatorial Aspects of Covers of Groups by Cosets or Subgroups, 2006.
  30. Curious Identities and Congruences involving Bernoulli Polynomials, 2006.
  31. Combinatorial Aspects of Szemeredi's Theorem, 2007.
  32. Additive Combinatorics and Latin Transversals, 2007.
  33. Sums of Squares and Triangular Numbers, and Rado Numbers for Linear Equations, 2007.
  34. An Additive Theorem Related to Latin Transversals, 2007.
  35. Some Famous Problems and Related Results in Combinatorial Number Theory, 2007.
  36. Groups in Combinatorial Number Theory, 2007.
  37. Various Extensions of Some Basic Results in Combinatorial Number Theory, 2008.
  38. Recent Problems and Results Involving Binomial Coefficients, 2008.
  39. An Extremal Problem on Covers of Abelian Groups, 2008.
  40. Study Covers of Groups via Characters and Number Theory, 2008.
  41. On Representations of Integers Involving Triangular Numbers, 2008.
  42. Some New Conjectures Involving Primes, 2009.
  43. Problems and Results in Additive Combinatorics, 2009.
  44. Polygonal Numbers, Primes and Ternary Quadratic Forms, 2009.
  45. Combinatorial Number Theory in China, 2009.
  46. On the DKSS Technique and the DKSS Conjecture, 2010.
  47. Some Sophisticated Applications of the Combinatorial Nullstellensatz, 2010.
  48. Conjectures and Results on Super Congruences, 2010.
  49. Conjectures for Super Congruences and Series for π and Other Constants (45 minutes version), 2010.
  50. Conjectures for Super Congruences and Series for π and Other Constants (30 minutes version), 2010.
  51. Super Congruences involving Binomial Coefficients and New Series for some Famous Constants, 2010.
  52. Arithmetic Properties of Combinatorial Quantities, 2010.
  53. On Divisibility concerning Binomial Coefficients, 2010.
  54. Correspondence between Series and Congruences, 2010.
  55. On Weighted Extension of the Erdos-Heilbronn Conjecture, 2011.
  56. On the DKSS Conjecture for Finite Abelian Groups, 2011.
  57. Number Theory behind Series for Powers of π, 2011.
  58. Conjectures and Results on x2 mod p2 with 4p=x2+dy2, 2011.
  59. Combinatorial Congruences via the Zeilberger Algorithm, 2011.
  60. On Arithmetic Properties of Bell Numbers, Delannoy Numbers and Schroder Numbers, 2011.
  61. Conjectures and Results on Generalized Trinomial Coefficients and Motzkin Numbers, 2011.
  62. Some Sophisticated Congruences involving Fibonacci Numbers, 2011.
  63. Some of my Number-theoretic Conjectures and related Progress, 2011.
  64. Historical Remarks on my Conjectural Congruences and Series for 1/π, 2011.
  65. Primes from the Viewpoint of Combinatorics, 2012.
  66. p-adic Congruences Motivated by Series, 2012.
  67. Various New Observations about Primes, 2012.
  68. Conjectures involving Arithmetical Sequences, 2012.
  69. The Riddle of Primes, 2012.
  70. Apery Numbers, Franel Numbers and Binary Quadratic Forms, 2013.
  71. Some New Representation Problems involving Primes, 2013.
  72. Combinatorial Congruences via Zeilberger's Algorithm and Trees with Prime Vertices, 2013.
  73. Some Open Combinatorial Congruences, 2013.
  74. Write n = k + m with f(k,m) Prime, 2013.
  75. Congruences for Franel Numbers, 2014.
  76. Problems on Combinatorial Properties of Primes, 2014.
  77. On Some Arithmetic Functions, 2014.
  78. Towards the Twin Prime Conjecture, 2014.
  79. Supercongruences Motivated by e, 2014.
  80. On Generalized Central Trinomial Coefficients, 2015.
  81. On Universal Sums involving Polygonal Numbers, 2015.
  82. New Divisibility Results on Certain Sums of Binomial Coefficients, 2015.
  83. On g_n(x)=sum_{k=0}^n binom(n,k)^2*binom(2k,k)*x^k and Related Topics, 2015.
  84. On Primes in Arithmetic Progressions, 2015.
  85. Some New Problems and Results in Combinatorial and Additive Number Theory, 2016.
  86. Some New Diophantine Problems, 2016.
  87. A New Result in Combinatorial Number Theory, 2016.
  88. Combinatorial Quantities and Arithmetic Means, 2016.
  89. Refining Lagrange's Four-square Theorem, 2016.
  90. The 1-3-5-Conjecture and Related Topics, 2016-2017.
  91. Restricted Sums of Three or Four Squares, 2017.
  92. Further Results on Hilbert's Tenth Problem, 2017.
  93. On Hilbert's Tenth Problem, 2017.

Selected Photographs
  1. at Venice (Venezia), Florence (Firenze), Genova, Rome (Roma), Trieste (2004)
  2. at Vienna (Wien), Graz (2004), Lyon, Bordeaux, Pacific beach (2005), Osaka (2008)
  3. at MIT [1,2] and University of California at Irvine [1,2,3] (2006)
  4. at Univ. of Wisconsin at Madison and Univ. of Illinois at Urbana Champaign (2006)
  5. at Stanford University [1, 2, 3, 4, 5] (2010)
  6. with Prof. M. Agrawal (famous for the AKS primality test) at ICTP, Trieste (2004)
  7. with Prof. R. Schoof (famous for Schoof's algorithm) at Rome (2004)
  8. with Prof. A. Perelli at Genova (2004)
  9. with Prof. A. Geroldinger and his wife at Graz (2004)
  10. with Prof. R. P. Stanley, R. A. Askey and J. Zeng at Tianjin (2004)
  11. with Prof. Y. Bilu at Bordeaux (2005)
  12. with Prof. J. Zeng and Prof. J. L. Nicolas at Institute of Camille Jordan (2005)
  13. Photographs at the Integers Conference 2005 (West Georgia Univ., Oct. 27-30):
    Prof. R. L. Graham, the Graham couple, Prof. C. Pomerance,
    Prof. M. B. Nathanson, Prof. Fan Chung and M. B. Nathanson, Prof. B. Landman;
    with Prof. R.L. Graham and Fan Chung, C. Pomerance and S. Wagstaff,
    with Prof. M. B. Nathanson, H. Diamond and D. Goldston, B. Landman, S. Milne,
    with Prof. A. Bialostocki, T. Brown, K. O'Bryant and R. L. Jin, V. F. Lev.
  14. with Prof. R. Askey, K. Ono and T. H. Yang at Univ. of Wisconsin at Madison (2006)
  15. with Prof. H. Halberstam and P. T. Bateman at UI Urbana-Champaign (2006)
  16. with Prof. R. P. Stanley at MIT (Massachusetts Institute of Technology) (2006),
    with Prof. B. Green (famous for the Green-Tao theorem) at his MIT office (2006)
  17. with Prof. K. Rubin and D. Wan at Univ. of California at Irvine (2006)
  18. with Prof. B. Berndt at Nanjing University (2006)
  19. with Prof. A. Schinzel at Weihai (Shandong Province, China) (2006)
  20. with Prof. W. Konen at Osaka (Japan) (2008)
  21. with Prof. R. Schoof and Z. H. Sun in Xuanwu Park at Nanjing (2008)
  22. with Prof. M. Waldschmidt in India (2010)
  23. with Prof. S. D. Adhikari and Dr. D. J. Grynkiewicz in India (2010)
  24. with Prof. Wen-Ching Li, Don Zagier and his wife in Taiwan (2010)
  25. Pictures at the 1st Workshop on Number Theory, Combinatorics and their Interactions (Nanjing Univ., August 10-12, 2007):
    A Collective Photograph of Participants, Prof. Wen-Ching Winnie Li [1, 2, 3];
    Prof. Qin Yue, Zhi-Wei Sun, Keqin Feng, Wen-Ching W. Li, Daqing Wan and Yong-Gao Chen;
    W. C. Li, Z. W. Sun and D. Wan, W. C. Li and Z. W. Sun, W. C. Li and D. Wan.

   Curriculum Vitae of Zhi-Wei Sun
 Zhi-Wei Sun has the copyright of those unpublished materials at this website.
    The copyright of each published or accepted paper is held by the corresponding publisher.