¡¡¡¡¡¡¡¡¡¡¡¡
¡¡ ¡¡ visits since April 10, 2002 ¡¡¡¡¡¡ ¡¡¡¡¡¡ ¡¡¡¡¡¡

 ¡¡¡¡¡¡¡¡¡¡¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡Zhi-Wei Sun ¡¡¡¡¡¡¡¡¡¡¡¡Department of Mathematics ¡¡¡¡¡¡¡¡¡¡¡¡Nanjing University ¡¡¡¡¡¡¡¡¡¡¡¡Nanjing 210093 ¡¡¡¡¡¡¡¡¡¡¡¡People's Republic of China ¡¡¡¡¡¡¡¡¡¡¡¡E-mail: zwsun@nju.edu.cn ¡¡¡¡¡¡¡¡¡¡¡¡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.

¡¡¡¡¡¡ Editor-in-Chief of Journal of Combinatorics and Number Theory, 2009--.
¡¡¡¡¡¡¡¡ You may submit your paper by sending the pdf file to zwsun@nju.edu.cn
¡¡¡¡¡¡¡¡ 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 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
etc.

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

¡¡¡¡ Any natural number n can be written as the sum of squares of four nonnegative integers w, x, y and z such that x+3y+5z is also a square. [I have verified this for n up to 2*106.]

My Conjecture on Unit Fractions involving Primes

¡¡¡¡ 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 Representations involving Cubes

¡¡¡¡ 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 involving the n-th Prime

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

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 60 Open Problems on Combinatorial Properties of Primes

My Conjecture on the Prime-Counting Function

¡¡(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"

¡¡¡¡ 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 involving Primes and Powers of 2

¡¡¡¡ 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 1.6*106.]

My Conjecture on Sums of Primes and Numbers of the Form 2k-k

¡¡¡¡ Any integer n>3 can be written in the form p + (2k - k) + (2m - m), where p is a prime, and k and m are positive integers. [This has been verified for n up to 1010.]

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 Unification of Goldbach's Conjecture and the Twin Prime Conjecture

¡¡¡¡ 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 related to

¡¡¡¡ 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 Conjecture on Twin Primes and Sexy Primes

¡¡¡¡ Every n = 12, 13, ... can be written as p+q with p, p+6, 6q-1 and 6q+1 all prime. [I have verified this for n up to 1,000,000,000.]

My Curious Conjecture on Primes

¡¡¡¡ Each n = 2, 3, ... can be written as x2 + y, where x and y are nonnegative integers with 2y2 - 1 prime.

My Conjecture on Prime Differences (see also arXiv:1211.1588 for more conjectures on primes)

¡¡¡¡ Any integer n>7 can be written as p+q, where q is a positive integer, and p and 2pq+1 are primes. In general, for each m=0,1,2,..., any sufficiently large integer n can be written as x+y, where x and y are positive integers with x-m, x+m and 2xy+1 all prime. [I have verified the first assertion for n up to 1,000,000,000. The second assertion implies that for any positive even integer d there are infinitely many prime pairs {p,q} with p-q=d.]

My Conjectures on Representations via Sparse Primes

¡¡¡¡ Each integer n>3 can be written as p+q with p, 2p2-1 and 2q2-1 all prime, where q is a positive integer. (See OEIS A230351.)

My Conjecture on Primes of the Form an+b

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 15 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.)

Publications
¡¡¡¡¡¡
¡¡¡¡¡¡Papers Indexed in SCI or SCI-E
¡¡¡¡¡¡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

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.

¡¡¡¡¡¡¡¡
¡¡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.