In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv3\pmod4$, we show that $$(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$$ and $$(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$$ as conjectured by Sun, where $$(c,d)_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{1\le i,j\le n-1}$$ and $$[c,d]_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{0\le i,j\le n-1}$$ with $(\frac{\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\equiv5\pmod{12}$, and $[5,5]_p=0$ for any prime $p\equiv 13,17\pmod{20}$, which were also conjectured by Sun. Our proofs involve character sums over finite fields.
We study averages involving the Jacobi quadratic symbol $(\frac{n}{m})$ in regions where the product $mn$ is bounded by a large parameter. We show that these averages exhibit cancellation whenever the summation is restricted to square-free integers bounded away from the axes.
The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $a\equiv1\pmod4$. Then for the determinant $$S(1,p):={\rm det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
In 1989, Ming Luo \cite{L2} showed that the Fibonacci number $U_n$ is Triangular if and only if $n=\pm1,2,4,8,10$. For this, he established a Jacobi Symbol Criterion. Moreover, he observed that this problem is equivalent to finding all integer points on two elliptic curves. In this paper, we prove a Jacobi Symbol Criterion for more general families of binary recurrences. In addition, applying the criterion and elementary methods, we determine all integer points on the elliptic curves $y^2=5x^2(x+3)^2+4(-1)^n$.
Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for all $a\in \mathbb Z/p\mathbb Z$. We identify various symmetry properties of sequential matrices over $\mathbb Z/(n^2+1)\mathbb Z$ regardless of whether $n^2+1$ is prime. We deduce from these results a collection of symmetries involving Jacobi symbol modulo $n^2+1$ which generalize our above observation on the Legendre symbol.
In this paper we mainly focus on some determinants with Legendre symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0<k<\frac p4:\ (\frac kp)=-1\}|}&\text{if}\ p\equiv1\pmod4, \\(-1)^{\lfloor(p+1)/8\rfloor}&\text{if}\ p\equiv3\pmod4,\end{cases}$$ where $$S(d,p)=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}$$ and $$W_p=\det\left[\left(\frac{i^2-((p-1)/2)!j}p\right)\right]_{0\le i,j\le(p-1)/2}.$$ We …
We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.
On a determinant involving linear combinations of Legendre symbols
School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University（西安利物浦大学数学与物理学院）；School of Mathematics, Nanjing University（南京大学数学系）；School of Physical and Mathematical Sciences, Nanjing Tech University（南京工业大学物理与数学科学学院）
In a recent work, the present author generalized a fundamental result of Gauss related to quadratic reciprocity, and also showed that the above result of Gauss is equivalent to a special case of a well-known result of Sylvester related to the Frobenius coin problem. In this note, we use this equivalence to show that the above generalization of the result of Gauss naturally leads to an interesting generalization of the result of Sylvester. To be precise, for given positive coprime integers $a$ and $b$, and for a family of values of $k$ in the interval $0 \leq k < (a-1)(b-1)$, we find the number…
We study the Kronecker symbol $\left(\frac st\right)$ for the sequence of the convergents $s/t$ of a purely periodic continued fraction expansion. Whereas the corresponding sequence of Jacobi symbols is always periodic, it turns out that the sequence of Kronecker symbols may be aperiodic. Our main result describes the period length in the periodic case in terms of the period length of the sequence of Jacobi symbols and gives a necessary and sufficient condition for the occurrence of the aperiodic case.
In this paper, by the tools of circulant matrices and hyperelliptic curves over finite fields, we study some arithmetic properties of certain determinants involving the Legendre symbols and $k$-th residues.
The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n) log n), using Schönhage's fast continued fraction algorithm combined with an identity due to Gauss. We give a different O(M(n) log n) algorithm based on the binary recursive gcd algorithm of Stehlé and Zimmermann. Our implementation - which to our knowledge is the first to run in time O(M(n) log n) - is faster than GMP's quadratic implementation for inputs larger than about 10000 decimal digits.
In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where $S_c(…
For $A\subseteq\mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$ whose $A$-weighted sum is zero. When $A$ is the set of all units in $\mathbb Z_n$, we determine the value of $E_A(n)$ and values of two related constants $C_A(n)$ and $D_A(n)$. We also characterize all sequences of length $E_A(n)-1$ in $\mathbb Z_n$ which do not have any $A$-weighted zero-sum subsequence of length $n$ when $n$ is a power of 2.
For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted zero-sum constant $C_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence of consecutive terms. A sequence of length $C_A(n)-1$ in $\mathbb Z_n$ which does not have any $A$-weighted zero-sum subsequence of consecutive terms will be called a $C$-extremal sequence for $A$. Let $\big(\frac{x}{n}\big)$ denote the Jacobi symbol of $x\in\mathbb Z_n$. We characterize the $C$-extremal sequences for the weight-set $S(n)=\big\{\,x\in U(n):\big(\f…
Littlewood raised the question of how slowly the L_4 norm ||f||_4 of a Littlewood polynomial f (having all coefficients in {-1,+1}) of degree n-1 can grow with n. We consider such polynomials for odd square-free n, where ϕ(n) coefficients are determined by the Jacobi symbol, but the remaining coefficients can be freely chosen. When n is prime, these polynomials have the smallest known asymptotic value of the normalised L_4 norm ||f||_4/||f||_2 among all Littlewood polynomials, namely (7/6)^{1/4}. When n is not prime, our results show that the normalised L_4 norm varies considerably according t…
The Hankel determinants of a given power series $f$ can be evaluated by using the Jacobi continued fraction expansion of $f$. However the existence of the Jacobi continued fraction needs that all Hankel determinants of $f$ are nonzero. We introduce {\it Hankel continued fraction}, whose existene and unicity are guaranteed without any condition for the power series $f$. The Hankel determinants can also be evaluated by using the Hankel continued fraction. It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic. We prove a similar result for …
