For fixed m and a, we give an explicit description of those subsets of ${\mathbb F}_{q}$, q odd, for which both x and $mx+a$ are quadratic residues (and other combinations). These results extend and refine results that date back to Gauss.

We prove that the class of all the rings $\mathbb {Z}/m\mathbb {Z}$ for all $m>1$ is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles $\mathbb {A}_{\mathbb {Q}}$ of $\mathbb {Q}$.

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^{n}-1$. We say that the extension $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$-primitive elements property if, for every $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$ such that $\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$, there exists some $x\in \mathbb{F}_{q}$ such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$ has multiplicative order $(q^{n}-1)/r$. We prove that, for sufficiently large prime powers $q$, $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$-primitive elements. We also discuss the (weaker) translate property for extensions.

A Ducci sequence is a sequence of integer $n$-tuples obtained by iterating the map

$$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$

$P(n)$

$n$

$P(n)$

$n$

Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, with $C(f)$ exceptions, the solutions $(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where $c$ is a constant and ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline {\mathbb F}_p^*$. Moreover, for most $p\lt N$, $N$ being a large number, we prove that, with $C(f)$ exceptions, ${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where $\epsilon (p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty $.

We consider the equation over a finite field q of q elements, with variables from arbitrary sets . The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that if for some absolute constant C > 0, then above equation has a solution for any λ ∈ q*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.

The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree $n$ over any finite field with any coefficient arbitrarily prescribed. This has recently been proved whenever $n \geq 9$. It is also known to be true when $n \leq 3$. We show that there exists a primitive polynomial of any degree $n\geq 4$ over any finite field with its second coefficient (i.e., that of $x^{n-2}$) arbitrarily prescribed. In particular, this establishes the HMPC when $n=4$. The lone exception is the absence of a primitive polynomial of the form $x^4+a_1x^3 +x^2+a_3x+1$ over the binary field. For $n \geq 6$ we prove a stronger result, namely that the primitive polynomial may also have its constant term prescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs.