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
¡¡¡¡¡¡1980.9--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 book New Conjectures in Number Theory and Combinatorics
(ÊýÂÛÓë×éºÏÖеÄвÂÏë) (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 a2 + b2 + 3c + 5d
with a, b, c, d nonnegative integers. [This has been verified for n up to 2.4*1011.]
My Four-square Conjecture with
$2500 Prize for the First Proof
¡¡¡¡ Every n = 2,3,... can be written as x2 + y2
+ (2a3b)2
+ (2c5d)2,
where x,y,a,b,c,d are nonnegative integers. [This has been verified for n up to 1.6*1011 by Giovanni Resta.]
My 2-4-6-8
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*1012 by Yaakov Baruch.]
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
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 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
pk,...,pn
(k < n) not exceeding 2m+2.2*sqrt(m) such that
m = pn - pn-1 + ... + (-1)n-k pk, where pj denotes the j-th prime.
[This has been verified for m up to 109 by Chang Zhang.]
My Conjecture on Primitive Roots of the Form x2+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 g-1 an integer square such that g is a primitive root modulo p. [I verified this for all primes below 107. Later, C. Greathouse extended the verification to all primes below 1010.]
My
1680-Conjecture 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
x4 + 1680y3z is a square. [This has been verified for n up to 108 by Qing-Hu Hou.]
My
Conjecture on the Representation n = x4 + y3 + z2 + 2k
with
$234 Prize (see also Conjecture 6.1(i) of this paper)
¡¡¡¡ Each n = 2,3,... can be written as x4 + y3 + z2 + 2k with x,y,z nonnegative integers and k a positive integer.
[This has been verified by Qing-Hu Hou for n up to 109.]
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 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 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 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.
[This has been verified for m up to 4*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 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 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 a1,
..., an of all the elements of A such that a1+a2,
..., an-1+an, an+a1
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 Hankel-type 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 (2010-2014) and
My 117 New Conjectural Series for Powers of π (2019)
¡¡¡¡ 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(40k+13)Tk(4,1)Tk(1,-1)2
/(-50)k
= 55*sqrt(15)/(9π),
¡¡¡¡¡¡¡¡ ∑k ≥ 0(1435k+113)Tk(7,1)Tk(10,10)2
/3240k
= 1452*sqrt(5)/π,
¡¡¡¡¡¡¡¡ ∑k>0(28k2-18k+3)(-64)k
/(k5C(2k,k)4C(3k,k))
= -14∑n>0 1/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)2∑k ≥ 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)2∑k ≥ 0 C(n,k)2C(2k,k)C(2(n-k),n-k)
= 24/π2.
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
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.
[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+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.)
|