83,166 research outputs found

    Consecutive primes and Legendre symbols

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    Let mm be any positive integer and let Ξ΄1,Ξ΄2∈{1,βˆ’1}\delta_1,\delta_2\in\{1,-1\}. We show that for some constanst Cm>0C_m>0 there are infinitely many integers n>1n>1 with pn+mβˆ’pn≀Cmp_{n+m}-p_n\le C_m such that (pn+ipn+j)=Ξ΄1Β andΒ (pn+jpn+i)=Ξ΄2\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=\delta_2 for all 0≀i<j≀m0\le i<j\le m, where pkp_k denotes the kk-th prime, and (β‹…p)(\frac {\cdot}p) denotes the Legendre symbol for any odd prime pp. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers nn such that pn+ip_{n+i} is a primitive root modulo pn+jp_{n+j} for any distinct ii and jj among 0,1,…,m0,1,\ldots,m.Comment: 12 pages, final published versio

    Evaluations of some Toeplitz-type determinants

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    In this paper we evaluate some Toeplitz-type determinants. Let n>1n>1 be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+\delta_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\ \det{[|j-k|+\delta_{jk}]_{1\leq j,k\leq n}}&= \begin{cases} \frac{1+(-1)^{(n-1)/2}n}{2}&\text{if}\ 2\nmid n,\\ \frac{1+(-1)^{n/2}}{2}&\text{if}\ 2\mid n, \end{cases} \end{align*} where Ξ΄jk\delta_{jk} is the Kronecker delta. For complex numbers a,b,ca,b,c with b=ΜΈ0b\not=0 and a2=ΜΈ4ba^2\not=4b, and the sequence (wm)m∈Z(w_m)_{m\in\mathbb Z} with wk+1=awkβˆ’bwkβˆ’1w_{k+1}=aw_k-bw_{k-1} for all k∈Zk\in\mathbb Z, we establish the identity det⁑[wjβˆ’k+cΞ΄jk]1≀j,k≀n=cn+cnβˆ’1nw0+cnβˆ’2(w12βˆ’aw0w1+bw02)un2b1βˆ’nβˆ’n2a2βˆ’4b,\det[w_{j-k}+c\delta_{jk}]_{1\le j,k\le n} =c^n+c^{n-1}nw_0+c^{n-2}(w_1^2-aw_0w_1+bw_0^2)\frac{u_n^2b^{1-n}-n^2}{a^2-4b}, where u0=0u_0=0, u1=1u_1=1 and uk+1=aukβˆ’bukβˆ’1u_{k+1}=au_k-bu_{k-1} for all k=1,2,…k=1,2,\ldots.Comment: 22 pages.Add parts (ii) and (iii) of Theorem 1.
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