Initial day:
July 31, 2001 ¡¡¡¡¡¡¡¡
Last modified:
July 19, 2016


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








Research Interests ¡¡¡¡¡¡ Number Theory (especially Combinatorial Number Theory), ¡¡¡¡¡¡ Combinatorics, Group Theory, Mathematical Logic. Academic Service ¡¡¡¡¡¡ EditorinChief 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 ¡¡¡¡¡¡1980.91983.7 The High Middle School Attached to Nanjing Normal Univ. ¡¡¡¡¡¡1983.91992.6 Department of Mathematics, Nanjing University ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡¡ (UndergraduatePh. D. Candidate; B. Sc. 1987, Ph. D. 1992) ¡¡¡¡¡¡1992.7 ¡¡¡¡¡¡ Teacher in Department of Mathematics, Nanjing University ¡¡¡¡¡¡1994.41998.3 Associate Professor in Math. ¡¡¡¡¡¡1998.4 ¡¡¡¡¡¡ Full Professor in Math. ¡¡¡¡¡¡1999.11 ¡¡¡¡¡¡ Supervisor of Ph. D. students One of My Conjectures refining Lagrange's Foursquare Theorem (see also OEIS A271518£© ¡¡¡¡ 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*10^{6}.] 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*10^{7}.] My Conjecture involving the nth Prime ¡¡¡¡ For each m = 1,2,3, ... there is a positive integer n such that m + n divides p_{m} + p_{n}, where p_{k} denotes the kth prime. Moreover, we may require that n < m*(m1) if m > 2. [I have verified this for m up to 10^{5}, 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*10^{9}.] 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 g1 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 10^{7} and 2*10^{7} respectively. Later, C. Greathouse extended the verification of the first assertion in the conjecture to all primes below 10^{10}.] My 60 Open Problems on Combinatorial Properties of Primes My Conjecture on the PrimeCounting 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 10^{7}. 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*10^{5}. See OEIS A238902 and OEIS A239884.] ¡¡(iii) For each integer n>2, π(np) is a square for some prime p < n. [I have verified this for n up to 5*10^{8}. 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 jth prime. [I have verified this for n up to 10^{9}.] 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 2^{k} + m is prime. [This has been verified for n up to 1.6*10^{6}.] My Conjecture on Sums of Primes and Numbers of the Form 2^{k}k ¡¡¡¡ Any integer n>3 can be written in the form p + (2^{k}  k) + (2^{m}  m), where p is a prime, and k and m are positive integers. [This has been verified for n up to 10^{10}.] 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 p_{n+1} is just the least positive integer m such that 2s_{k}^{2} (k=1,...,n) are pairwise distinct modulo m, where s_{k} = p_{k}p_{k1}+...+(1)^{k1}p_{1}. [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 p_{k},...,p_{n} (k < n) not exceeding 2m+2.2*sqrt(m) such that m = p_{n}p_{n1}+...+(1)^{nk}p_{k}. [I have verified this for m up to 10^{5}.] 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 nth prime. My Conjecture related to Bertrand's Postulate (see also A185636 and A204065 in OEIS) ¡¡¡¡ Let n be any positive integer. Then, for some k=0,...n, both n+k and n+k^{2} 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, 6q1 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 x^{2} + y, where x and y are nonnegative integers with 2y^{2}  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 xm, 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 pq=d.] My Conjectures on Representations via Sparse Primes ¡¡¡¡ Each integer n>3 can be written as p+q with p, 2p^{2}1 and 2q^{2}1 all prime, where q is a positive integer. (See OEIS A230351.) My Conjecture on Primes of the Form a^{n}+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 a_{1}, ..., a_{n} of all the elements of A such that a_{1}+a_{2}+a_{3}, ..., a_{n1}+a_{n}+a_{1}, a_{n}+a_{1}+a_{2} are pairwise distinct. [We have proved this for any torsionfree abelian group G, see also A228772 in OEIS.] My 15 Conjectures on Determinants (see also arXiv:1308.2900 and Three mysterious conjectures on Hankeltype 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!(nk)!) and let T_{n}(b,c) denote the coefficient of x^{n} in (x^{2}+bx+c)^{n}. Then ¡¡¡¡¡¡¡¡ ∑_{k ≥ 0} (126k+31)T_{k}(22,21^{2})^{3}/(80)^{3k} = 880*sqrt(5)/(21π), ¡¡¡¡¡¡¡¡ ∑_{k ≥ 0} (24k+5)C(2k,k)T_{k}(4,9)^{2}/28^{2k} = 49(sqrt(3)+sqrt(6))/(9π), ¡¡¡¡¡¡¡¡ ∑_{k ≥ 0}(2800512k+435257)C(2k,k)T_{k}(73,576)^{2} /434^{2k} = 10406669/(2sqrt(6)π), ¡¡¡¡¡¡¡¡ ∑_{k>0}(28k^{2}18k+3)(64)^{k} /(k^{5}C(2k,k)^{4}C(3k,k)) = 14∑_{n>0}1/n^{3}. ¡¡¡¡¡¡¡¡ ∑_{n ≥ 0}(28n+5)24^{2n} C(2n,n)∑_{k ≥ 0 }5^{k }C(2k,k)^{2}C(2(nk),nk)^{2}/C(n,k) = 9(sqrt(2)+2)/π. ¡¡¡¡¡¡¡¡ ∑_{n ≥ 0 }(18n^{2}+7n+1)(128)^{n} C(2n,n)^{2}∑_{k ≥ 0 }C(1/4,k)^{2}C(3/4,nk)^{2} = 4*sqrt(2)/π^{2}. ¡¡¡¡¡¡¡¡ ∑_{n ≥ 0}(40n^{2}+26n+5)(256)^{n} C(2n,n)^{2}∑_{k ≥ 0 }C(n,k)^{2}C(2k,k)C(2(nk),nk) = 24/π^{2}. My Hypothesis on the Parities of Ω(n)n (see also a public message and arXiv:1204.6689) ¡¡¡¡ We have {n ≤ x: nΩ(n) is even} > {n ≤ x: nΩ(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)^{nΩ(n)} > sqrt(x) for any x ≥ 325. [I have shown that the hypothesis implies the Riemann Hypothesis, and verified it for x up to 10^{11}.] My Conjecture on Sums of Primes and Triangular Numbers ¡¡¡¡ Each natural number not equal to 216 can be written in the form p+T_{x} , where p is 0 or a prime, and T_{x}=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 2^{a}p +T_{x} , where p is either zero or a prime congruent to r modulo 2^{b}. My Conjecture on Sums of Polygonal Numbers ¡¡¡¡ For each integer m>2, any natural number n can be expressed as p_{m+1}(x_{1}) + p_{m+2}(x_{2}) + p_{m+3}(x_{3}) + r with x_{1},x_{2},x_{3} nonnegative integers and r among 0,...,m3, where p_{k}(x)=(k2)x(x1)/2+x (x=0,1,2,...) are kgonal 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 a_{1}G_{1} , ..., a_{k}G_{k} (k>1) be finitely many pairwise disjoint left cosets in a group G with all the indices [G:G_{i}] finite. Then, for some distinct i and j the greatest common divisor of [G:G_{i}] and [G:G_{j}] is at least k. My Conjecture on Covers of Groups ¡¡¡¡ Let a_{1}G_{1} , ..., a_{k}G_{k} be finitely many left cosets in a group G which cover all the elements of G at least m>0 times with a_{j}G_{j} irredundant. Then k is at least m+f([G:G_{j}]), where f(1)=0 and f(p_{1} ... p_{r}) =(p_{1}1) + ... +(p_{r}1) for any primes p_{1} , ..., p_{r} . My Conjecture on Linear Extension of the ErdosHeilbronn Conjecture RedmondSun Conjecture (in PlanetMath.) 



Publications  
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¡¡¡¡¡¡Papers Indexed in SCI or SCIE  Papers Listed by Field  
¡¡¡¡¡¡Recent Publications (2008)  Preprints on arXiv  
¡¡¡¡¡¡Publications during 20002007  Publications during 19871999  
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Other Information  
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¡¡¡¡¡¡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 WallSunSun Prime [1, 2, 3]  
¡¡¡¡¡¡Covers, Sumsets and Zerosums  Link to the useful Number Theory Web  
¡¡¡¡¡¡Mixed Sums of Primes and Other Terms  Articles on arXiv: Combinatorics, Number Theory  
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Invited Lectures in Mathematics
 


Selected Photographs
 


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¡¡¡¡¡¡ The copyright of each published or accepted paper is held by the corresponding publisher.
