Supercongruences involving dual sequences

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom nk(-1)^k\binom xk\binom{-1-x}k.$$ For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{k=0}^{p-1}(\pm1)^kd_k(x)^2$ and $\sum_{k=0}^{p-1}(2k+1)d_k(x)^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\equiv(-1)^{\langle x\rangle_p}\pmod{p^2},$$ where $\langle x\rangle_p$ denotes the least nonnegative integer $r$ with $x\equiv r\pmod p$. For any prime $p>3$ and $p$-adic integer $x$, we determine $\sum_{k=0}^{p-1}s_k(x)^2$ modulo $p^2$ (or $p^3$ if $x\in\{0,\ldots,p-1\}$), and show that $$\sum_{k=0}^{p-1}(2k+1)s_k(x)^2\equiv0\pmod{p^2}.$$ We also pose several related conjectures.Comment: 35 pages, final published versio

New observations on primitive roots modulo primes

We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q-1}$ a primitive root modulo $p$. We also make related observations on quadratic nonresidues modulo primes and primitive prime divisors of some combinatorial sequences. For example, based on heuristic arguments we conjecture that for any prime $p>3$ there exists a Fibonacci number $F_k<p/2$ which is a quadratic nonresidue modulo $p$; this implies that there is a deterministic polynomial time algorithm to find square roots of quadratic residues modulo a prime $p>3$.Comment: 23 page

A result similar to Lagrange's theorem

Generalized octagonal numbers are those $p_8(x)=x(3x-2)$ with $x\in\mathbb Z$. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for $35$ triples $(b,c,d)$ with $1\le b\le c\le d$ (including $(2,3,4)$ and $(2,4,8)$), we prove that any nonnegative integer can be exprssed as $p_8(w)+bp_8(x)+cp_8(y)+dp_8(z)$ with $w,x,y,z\in\mathbb Z$. We also pose several conjectures for further research.Comment: 21 pages, final published versio

Quadratic residues and related permutations and identities

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ (with $\{k\}_p$ the least nonnegative residue of $k$ modulo $p$) is a permutation of $a_1,\ldots,a_{(p-1)/2}$, and we show that the sign of this permutation is $1$ or $(-1)^{(h(-p)+1)/2}$ according as $p\equiv3\pmod 8$ or $p\equiv7\pmod 8$, where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. To achieve this, we evaluate the product $\prod_{1\le j<k\le(p-1)/2}(\cot\pi j^2/p-\cot\pi k^2/p)$ via Dirichlet's class number formula and Galois theory. We also obtain some new identities for the sine and cosine functions; for example, we determine the exact value of $$\prod_{1\le j<k\le p-1}\cos\pi\frac{aj^2+bjk+ck^2}p$$ for any $a,b,c\in\mathbb Z$ with $ac(a+b+c)\not\equiv0\pmod p$.Comment: 36 pages, final published versio

An additive theorem and restricted sumsets

Let G be any additive abelian group with cyclic torsion subgroup, and let A, B and C be finite subsets of G with cardinality n>0. We show that there is a numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the elements of B and a numbering {c_i}_{i=1}^n of the elements of C, such that all the sums a_i+b_i+c_i (i=1,...,n) are distinct. Consequently, each subcube of the Latin cube formed by the Cayley addition table of Z/NZ contains a Latin transversal. This additive theorem can be further extended via restricted sumsets in a field

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<n^2p_n/3 for all n>30.Comment: 6 pages. Make Theorem 1.2 genera

On Motzkin numbers and central trinomial coefficients

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$ have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\frac2n\sum_{k=1}^n(2k+1)M_k^2\in\mathbb Z,$$ $$\frac{n^2(n^2-1)}6\,\bigg|\,\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$Comment: 21 page

On sums of primes and triangular numbers

We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number not equal to 216 can be written in the form p+T_x with x an integer and p a prime or zero; another conjecture asserts that any odd integer n>3 can be written in the form p+x(x+1) with p a prime and x a positive integer

On Snevily's conjecture and restricted sumsets

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every positive integer m\leq (k-1)/(n-1) there are more than (k-1)n-(m+1)n(n-1)/2 sets {a_1,...,a_n} such that a_1\in A_1,..., a_n\in A_n, and both a_i\not=a_j and ma_i+b_i\not=ma_j+b_j (or both ma_i\not=ma_j and a_i+b_i\not=a_j+b_j) for all 1\leq i<j\leq n. This extends a recent result of Dasgupta, K\'arolyi, Serra and Szegedy on Snevily's conjecture. Actually stronger results on sumsets with polynomial restrictions are obtained in this paper.Comment: 14 page

Arithmetic Properties of Periodic Maps

Let $\psi_1,...,\psi_k$ be periodic maps from $\Bbb Z$ to a field of characteristic p (where p is zero or a prime). Assume that positive integers $n_1,...,n_k$ not divisible by p are their periods respectively. We show that $\psi_1+...+\psi_k$ is constant if $\psi_1(x)+...+\psi_k(x)$ equals a constant for |S| consecutive integers x where S={r/n_s: r=0,...,n_s-1; s=1,...,k}. We also present some new results on finite systems of arithmetic sequences.Comment: 10 pages; accepted by Math. Res. Lett. Also available from http://pweb.nju.edu.cn/zwsu