Initial day:
August 2, 2001 ¡¡¡¡¡¡¡¡
Last modified:
October 5, 2022


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visits since 2012 ¡¡¡¡¡¡ ¡¡¡¡¡¡ ¡¡¡¡¡¡








Research Interests ¡¡¡¡¡¡ Number Theory (especially Combinatorial Number Theory), ¡¡¡¡¡¡ Combinatorics, Group Theory, Mathematical Logic. Academic Service ¡¡¡¡¡¡ Editorial Board Member of Electronic Research Archive, 2019. ¡¡¡¡¡¡ Reviewer for Zentralblatt Math., 2007. ¡¡¡¡¡¡ Reviewer for Mathematical Reviews, 1992. ¡¡¡¡¡¡ 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 etc. ¡¡¡¡¡¡ I refuse to referee papers for any open access journal which asks for page charges. 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 My paper Further results on Hilbert's Tenth Problem (based on my PhD thesis in 1992) My book New Conjectures in Number Theory and Combinatorics (ÊýÂÛÓë×éºÏÖÐµÄÐÂ²ÂÏë) (which collects 820 open conjectures posed by me) My book Modern Algebra (½üÊÀ´úÊý) £¨which is a textbook for undergraduate students) My Favorite Conjecture with $3500 (3500 US dollars) Prize for the First Proof (see OEIS A303389, A303540 and A303821 for similar conjectures) ¡¡¡¡ Any integer n > 1 can be written as a^{2} + b^{2} + 3^{c} + 5^{d} with a, b, c, d nonnegative integers. [This has been verified for n up to 2.4*10^{11}.] My Foursquare Conjecture with $2500 Prize for the First Proof ¡¡¡¡ Every n = 2,3,... can be written as x^{2} + y^{2} + (2^{a}3^{b})^{2} + (2^{c}5^{d})^{2}, where x,y,a,b,c,d are nonnegative integers. [This has been verified for n up to 1.6*10^{11} by Giovanni Resta.] My 2468 Conjecture with $2468 Prize for the First Proof ¡¡¡¡ Any positive integer n can be written as binom(w,2) + binom(x,4) + binom(y,6) + binom(z,8) with w,x,y,z integers greater than one. [This has been verified for n up to 2*10^{12} by Yaakov Baruch.] My 24Conjecture 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 10^{10} by QingHu 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 2^{k} + m is prime. [This has been verified for n up to 10^{7}.] My Conjecture on Alternating Sums of Consecutive Primes with $1000 Prize (see 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}, where p_{j} denotes the jth prime. [This has been verified for m up to 10^{9} by Chang Zhang.] 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 Primitive Roots of the Form x^{2}+1 with 2000 RMB Prize (see OEIS A239957, A241476 and Conj. 3.1 of this paper) ¡¡¡¡ For any prime p, there is an integer 0 < g < p with g1 an integer square such that g is a primitive root modulo p. [I verified this for all primes below 10^{7}. Later, C. Greathouse extended the verification to all primes below 10^{10}.] My 1680Conjecture with 1680 RMB Prize (see also OEIS A280831 and Conjecture 4.10(iv) of this published paper£© ¡¡¡¡ Any natural number n can be written as the sum of squares of four nonnegative integers x, y, z and w such that x^{4} + 1680y^{3}z is a square. [This has been verified for n up to 10^{8} by QingHu Hou.] My Conjecture on the Representation n = x^{4} + y^{3} + z^{2} + 2^{k} with $234 Prize (see also Conjecture 6.1(i) of this paper) ¡¡¡¡ Each n = 2,3,... can be written as x^{4} + y^{3} + z^{2} + 2^{k} with x,y,z nonnegative integers and k a positive integer. [This has been verified by QingHu Hou for n up to 10^{9}.] My Conjecture on Primes of the Form x^{2}+ny^{2} 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 x^{2}+ny^{2} are both prime. My Little 135 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 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+k^{2} 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 60 Open Problems on Combinatorial Properties of Primes My Conjecture on the PrimeCounting 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 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" (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 jth prime. [I have verified this for n up to 10^{9}.] 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 nth prime. My Conjecture involving the nth 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 p_{m} + p_{n}, where p_{k} denotes the kth prime. Moreover, we may require that n < m*(m1) if m > 2. [This has been verified for m up to 4*10^{5}.] 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 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*10^{7}.] My Conjecture on Representations involving Fourth Powers (see also Conjecture 1.2(iv) of this paper) ¡¡¡¡ Any nonnegative integer n can be written as x^{4} + 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 30 Conjectures in Additive Combinatorics ¡¡¡¡ Let G be an additive abelian group of odd order. Then, for any subset A of G with A=n>2 there is a numbering a_{1}, ..., a_{n} of all the elements of A such that a_{1}+a_{2}, ..., a_{n1}+a_{n}, a_{n}+a_{1} are pairwise distinct. [This has been verified for G<30 by Yuxuan Ji.] My Conjectures on Determinants (see also arXiv:1308.2900 and Three mysterious conjectures on Hankeltype determinants) My 100 Open Conjectures on Congruences £¨For the old preprint version on arXiv, see arXiv:0911.5665) My 234 Conjectural Series for Powers of π and Other Constants (20102014) and My 117 New Conjectural Series for Powers of π (2019) ¡¡¡¡ 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}(40k+13)T_{k}(4,1)T_{k}(1,1)^{2} /(50)^{k} = 55*sqrt(15)/(9π), ¡¡¡¡¡¡¡¡ ∑_{k ≥ 0}(1435k+113)T_{k}(7,1)T_{k}(10,10)^{2} /3240^{k} = 1452*sqrt(5)/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 Conjecture on Recurrence for Primes (see also Conj. 1.2 of this published paper and OEIS A181901) ¡¡¡¡ 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}. [This has been verified for n ≤ 300000.] 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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My Mathematical Lectures or Talks
 


Selected Photographs
 


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