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Thomas’ conjecture is, given monic polynomials $p_1,$ $...,p_d\in \mathbb{Z}\left[a\right]$ with $0\<deg{p}_1\<\cdots \<deg{p}_d$, then the Thue equation (over the rational integers) $$(X-p_1\left(a\right)Y)\cdots (X-p_d\left(a\right)Y)+Y^d=1$$ has only trivial solutions, provided $a\ge a_0$ with effective computable $a_0$. We consider a function field analogue of Thomas’ conjecture in case of degree $d=3$. Moreover we find a counterexample to Thomas’ conjecture for $d=3$.

Given a binary recurrence ${u_n}_{n\ge 0}$, we consider the Diophantine equation $u_{n_1}^{x_1}\cdots u_{n_L}^{x_L}=1$ with nonnegative integer unknowns $n_1,...,n_L$, where $n_i\ne n_j$ for 1 ≤ i < j ≤ L, $max|x_i|:1\le i\le L\le K$, and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

Let $(a_1,\cdots ,a_m)$ be an $m$-tuple of positive, pairwise distinct integers. If for all $1\le i<j\le m$ the prime divisors of $a_i{a}_j+1$ come from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In this note we estimate the number of $S$-Diophantine quadruples in terms of $\left|S\right|=r$.

All purely cubic fields such that their maximal order is generated by its units are determined.

Let g ≥ 2 be an integer and ${}_g\subset \mathbb{N}$ be the set of repdigits in base g. Let ${}_g$ be the set of Diophantine triples with values in ${}_g$; that is, ${}_g$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set ${}_g$. We prove effective finiteness results for the set ${}_g$.