Let m be any positive integer and let Ξ΄1β,Ξ΄2ββ{1,β1}. We
show that for some constanst Cmβ>0 there are infinitely many integers n>1
with pn+mββpnββ€Cmβ such that
(pn+jβpn+iββ)=Ξ΄1βΒ andΒ (pn+iβpn+jββ)=Ξ΄2β for all 0β€i<jβ€m,
where pkβ denotes the k-th prime, and (pβ β) denotes the
Legendre symbol for any odd prime p. We also prove that under the Generalized
Riemann Hypothesis there are infinitely many positive integers n such that
pn+iβ is a primitive root modulo pn+jβ for any distinct i and j
among 0,1,β¦,m.Comment: 12 pages, final published versio
In this paper we evaluate some Toeplitz-type determinants. Let n>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΄jkβ is the Kronecker delta. For complex numbers a,b,c with
bξ =0 and a2ξ =4b, and the sequence (wmβ)mβZβ with
wk+1β=awkββbwkβ1β for all kβZ, we establish the identity
det[wjβkβ+cΞ΄jkβ]1β€j,kβ€nβ=cn+cnβ1nw0β+cnβ2(w12ββaw0βw1β+bw02β)a2β4bun2βb1βnβn2β,
where u0β=0, u1β=1 and uk+1β=aukββbukβ1β for all k=1,2,β¦.Comment: 22 pages.Add parts (ii) and (iii) of Theorem 1.