We consider the Legendre quadratic forms$${\varphi }_{ab}(x,y,z)=a{x}^2+b{y}^2-z^2$$and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers $1\le a\le A,1\le b\le B$, for which the form ${\varphi }_{ab}$ has a non-trivial rational zero. Under certain mild conditions on the integers $a,b$, we are able to find the asymptotic formula for the number of such forms.

Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$. It is known that there are unique integers $A$, $B$, $C$, $D$ such that $$K=Q\left(\sqrt{A(D+B\sqrt{D})}\right),$$ where $$A\text{is}\text{squarefree}\text{and}\text{odd},D=B^2+C^2\text{is}\text{squarefree},B>0,C>0,GCD(A,D)=1.$$ The conductor $f\left(K\right)$ of $K$ is $f\left(K\right)=2^l\left|A\right|D$, where $$l=\left\{\begin{array}{c}3,\text{if}D\equiv 2(mod4)\text{or}D\equiv 1(mod4),B\equiv 1(mod2),\hfill \\ 2,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 3(mod4),\hfill \\ 0,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 1(mod4).\hfill \end{array}\right.$$ A simple proof of this formula for $f\left(K\right)$ is given, which uses the basic properties of quartic Gauss sums.

It is proved that the sequence $\left[n^c\right](n=1,2,\cdots )$ contains infinite squarefree integers whenever $1\<c\<\frac{61}{36}=1.6944\cdots$, which improves Rieger’s earlier range $1\<c\<1.5$.

Recently Garashuk and Lisonek evaluated Kloosterman sums K (a) modulo 4 over a finite field F3m in the case of even K (a). They posed it as an open problem to characterize elements a in F3m for which K (a) ≡ 1 (mod4) and K (a) ≡ 3 (mod4). In this paper, we will give an answer to this problem. The result allows us to count the number of elements a in F3m belonging to each of these two classes.