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Zhi-Wei Sun

Publications and source records attributed to Zhi-Wei Sun.

At least 163 records · Page 9Linked to original sources
arXiv2013

Fibonacci numbers modulo cubes of primes

Let $p$ be an odd prime. It is well known that $F_{p-(\frac p5)}\equiv 0\pmod{p}$, where $\{F_n\}_{n\ge0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p\not=5$ then we may determine $F_{p-(\frac p5)}$ mod $p^3$ in the following way: $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k}{(-16)^k}\equiv\left(\frac{p}5\right)\left(1+\frac{F_{p-(\frac {p}5)}}2\right)\pmod{p^3}.$$ We also use Lucas quotients to determine $\sum_{k=0}^{(p-1)/2}\binom{2k}k/m^k$ modulo $p^2$ for any integer $m\not\equiv0\pmod{p}$; in particular, we obtain $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k}{16^k}\equiv\left(\frac3{p}\right)\pmod{p^2}.$$ In addition, we pose three conjectures for further research.

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arXiv2013

On a sequence involving sums of primes

For $n=1,2,3,\ldots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\root n\of{S_n/n}\ (n=1,2,3,\ldots)$ is strictly decreasing, and moreover the sequence $a_{n+1}/a_n\ (n=10,11,\ldots)$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.

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arXiv2013

Conjectures involving arithmetical sequences

We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form $(\root n\of{a_n})_{n\ge 1}$ or the form $(\root{n+1}\of{a_{n+1}}/\root n\of{a_n})_{n\ge1}$, where $(a_n)_{n\ge 1}$ is a number-theoretic or combinatorial sequence of positive integers. This material might stimulate further research.

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arXiv2013

Arithmetic theory of harmonic numbers (II)

For $k=1,2,\ldots$ let $H_k$ denote the harmonic number $\sum_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have $$\sum_{k=1}^{p-1}\frac{H_k}{k2^k}\equiv\frac7{24}pB_{p-3}\pmod{p^2},\ \ \sum_{k=1}^{p-1}\frac{H_{k,2}}{k2^k}\equiv-\frac 38B_{p-3}\pmod{p},$$ and $$\sum_{k=1}^{p-1}\frac{H_{k,2n}^2}{k^{2n}}\equiv\frac{\binom{6n+1}{2n-1}+n}{6n+1}pB_{p-1-6n}\pmod{p^2}$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,\ldots$ are Bernoulli numbers, and $H_{k,m}:=\sum_{j=1}^k 1/j^m$.

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arXiv2013

Supercongruences involving products of two binomial coefficients

In this paper we deduce some new supercongruences modulo powers of a prime $p>3$. Let $d\in\{0,1,\ldots,(p-1)/2\}$. We show that $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{8^k}\equiv 0\ (\mbox{mod}\ p)\ \ \ \mbox{if}\ d\equiv \frac{p+1}2\ (\mbox{mod}\ 2),$$ and $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{16^k} \equiv\left(\frac{-1}p\right)+p^2\frac{(-1)^d}4E_{p-3}\left(d+\frac12\right)\pmod{p^3},$$ where $E_{p-3}(x)$ denotes the Euler polynomial of degree $p-3$, and $(-)$ stands for the Legendre symbol. The paper also contains some other results such as $$\sum_{k=0}^{p-1}k^{(1+(\frac{-1}p))/2}\frac{\binom{6k}{3k}\binom{3k}k}{864^k}\equiv0\pmod{p^2}.$$

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arXiv2013

Congruences for Franel numbers

The Franel numbers given by $f_n=\sum_{k=0}^n\binom{n}{k}^3$ ($n=0,1,2,\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We mainly establish for any prime $p>3$ the following congruences: \begin{align*}\sum_{k=0}^{p-1}(-1)^kf_k&\equiv\left(\frac p3\right)\ \ (\mbox{mod}\ p^2), \\ \sum_{k=0}^{p-1}(-1)^k\,kf_k&\equiv-\frac 23\left(\frac p3\right)\ \ (\mbox{mod}\ p^2), \\ \sum_{k=1}^{p-1}\frac{(-1)^k}kf_k &\equiv0\ \ (\mbox{mod}\ p^2), \\ \sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}f_k&\equiv0\ \ (\mbox{mod}\ p). \end{align*}

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arXiv2013

On functions taking only prime values

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the set of all prime numbers. For every n=4,5,... we prove that the least prime p>3n with 3|p-1 is just the least positive integer m such that 18k(3k-1) (k=1,...,n) are pairwise distinct modulo m. For d=4,6,12 and n=3,4,...., we prove that the least prime p>2n-2 with p=-1 (mod d) is the smallest integer m such that those (2k-1)^d for k=1,...,n are pairwise distinct modulo m. We also pose several challenging conjectures on primes. For example, we find a surprising recurrence for primes, namely, for every n=10,11,... 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 = sum_{j=1}^k(-1)^{k-j}p_j. We also conjecture that 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_{n-1}+...+(-1)^{n-k}p_k.

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arXiv2012

On irreducible polynomials over finite fields

For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence N_{n+1}(q)^{1/(n+1)}/N_n(q)^{1/n} (n>=5.835*10^{14}) is strictly decreasing. We also prove that if q>8 then N_{n+1}(q)/N_n(q) (n=1,2,3,...) is strictly increasing.

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arXiv2012

Some new inequalities for primes

For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k 30.

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arXiv2011

On weighted zero-sum sequences

Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that s_A(G) is at most $\lceil D(G)/|A|\rceil+exp(G)-1$ if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is essentially best possible. In the case A={1,-1}, we determine the asymptotic behavior of s_{1,-1}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s_{1,-1}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends to the infinity. Combined with a lower bound of $exp(G)+sum{i=1}{r}\lfloor\log_2 n_i\rfloor$, where $G=\Z_{n_1}\oplus...\oplus \Z_{n_r}$ with 1<n_1|... |n_r, this determines s_{1,-1}(G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems. Some additional more specific values and results related to s_{1,-1}(G) are also computed.

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arXiv2011

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $π^2$, $π^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=π^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$$

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arXiv2011

Linear extension of the Erdos-Heilbronn conjecture

The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements $a_1,...,a_n$ of F, the set {a_1x_1+...+a_nx_n: x_1,....,x_n are distinct elements of A} has cardinality at least min{p(F)-delta, n(|A|-n)+1}, where the additive order p(F) of the multiplicative identity of F is different from n+1, and delta=0,1 takes the value 1 if and only if n=2 and $a_1+a_2=0$. In this paper we prove this conjecture of Sun when $p(F)\geq n(3n-5)/2$. We also obtain a sharp lower bound for the cardinality of the restricted sumset {x_1+...+x_n: x_1\in A_1,...,x_n\in A_n, and P(x_1,...,x_n)\not=0}, where $A_1,...,A_n$ are finite subsets of a field F and $P(x_1,...,x_n)$ is a general polynomial over F.

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arXiv2011

Open Conjectures on Congruences

We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while conjectures in Part B have been recently confirmed. We hope that this material will interest number theorists and stimulate further research. Number theorists are welcome to work on those open conjectures; for some of them we offer prizes for the first correct proofs.

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arXiv2011

On Delannoy numbers and Schröder numbers

The n-th Delannoy number and the n-th Schröder number given by $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ and $S_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that $$\sum_{k=1}^{p-1}D_k/k^2=2(-1/p)E_{p-3} (mod p)$$ and $$\sum_{k=1}^{p-1}S_k/m^k=(m^2-6m+1)/(2m)*(1-((m^2-6m+1)/p) (mod p),$$ where (-) is the Legendre symbol, E_0,E_1,E_2,... are Euler numbers and m is any integer not divisible by p. We also conjecture that $\sum_{k=1}^{p-1}D_k^2/k^2=-2q_p(2)^2 (mod p)$, where $q_p(2)=(2^{p-1}-1)/p$.

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arXiv2011

On congruences related to central binomial coefficients

It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=π/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=π/3$. In this paper we obtain their p-adic analogues such as $$\sum_{p/2 3 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 and p=x^2+7y^2 with x,y integers, and $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=0 (mod p^2)$$ if (p/7)=-1, i.e., p=3,5,6 (mod 7).

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arXiv2011

On Apery numbers and generalized central trinomial coefficients

We decide to combine this preprint with two other ones (arXiv:1101.1946 and arXiv:1008.3887). The results on Apery numbers and polynomials will be added to the new version of arxiv:1101.1946, and those results on generalized trinomial coefficients will be added to the new version of arxiv:1008.3887

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