Uniform explicit Stewarts theorem on prime factors of linear recurrences

Stewart (2013) proved that the biggest prime divisor of the $n$th term of a Lucas sequence of integers grows quicker than $n$, answering famous questions of ErdH{o}s and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewarts result.

Non-trivial solutions to a linear equation in integers

For k>=3 let A subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A has O(N^{1/2-epsilon}) elements. This answers a question of I. Ruzsa.

Warnings Second Theorem with Resricted Variables

We present a restricted variable generalization of Warnings Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Brinks restricted variable generalization of Chevalleys Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warnings Second Theorem implies Chevalleys Theorem, our result implies Brinks Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems.

A note on Linniks Theorem on quadratic non-residues

We present a short, self-contained, and purely combinatorial proof of Linniks theorem: for any $varepsilon > 0$ there exists a constant $C_varepsilon$ such that for any $N$, there are at most $C_varepsilon$ primes $p leqslant N$ such that the least positive quadratic non-residue modulo $p$ exceeds $N^varepsilon$.

A structure theorem for product sets in extra special groups

Hegyvari and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $Bcdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.