Given an integer $N\geq 3$, we shall prove that for all primes $p\geq(N-2)^2 4^N$, there exists 𝑥 in $(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all squares (respectively, non-squares) modulo 𝑝. Similarly, for an integer $N\geq 2$, we prove that for all primes $p\geq \exp(2^{5.54N})$, there exists an element $x\in(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all generators of $(\mathbb{Z}/p\mathbb{Z})^∗$.

Let $\mathbb{F}$ be a field, $V$ a vector space of dimension $n$ over $\mathbb{F}$. Then the set of bilinear forms on $V$ forms a vector space of dimension $n^{2}$ over $\mathbb{F}$. For char $\mathbb{F} \neq 2$, if $T$ is an invertible linear map from $V$ onto $V$ then the set of $T$ -invariant bilinear forms, forms a subspace of this space of forms. In this paper, we compute the dimension of $T$ -invariant bilinear forms over $\mathbb{F}$. Also we investigate similar type of questions for the infinitesimally $T$ -invariant bilinear forms ($T$ -skew symmetric forms). Moreover, we discuss the existence of nondegenerate invariant (resp. infinitesimally invariant) bilinear forms.

In this paper, we find the 2$k$-th power mean of the inversion of $L$-functions with the weight of the general quartic Gauss sums. We establish the results with the help of Dirichlet characters and properties of classical Gauss sums. We also describe asymptotic behaviour for it.

Let $p$ be a prime $\equiv 1\pmod{p}$. If an integer $D$ with $(p,D)=1$ is an eleventh power nonresidue $\pmod{p}$, then $D^{(p-1)/11} \equiv \alpha\, \pmod{p}$ for some eleventh root of unity $\alpha(\not \equiv 1)\,\pmod{p}$. In this paper, we establish an explicit expression for $\alpha$ in terms of a particular solution of certain quadratic partition of $p$. Euler's criterion for eleventh power residues and nonresidues is given with explicit results for $D=2, 7,11$.