The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \mathbb{N}$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $min\{x,y,z\}>1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

A nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the...

Dati $m=2$ o $m=3$ numeri algebrici non nulli $\alpha =\left({\alpha }_1,\mathrm{\dots },{\alpha }_m\right)$ tali che ${\alpha }_j/{\alpha }_l$ non è una radice dell'unità per ogni $j\ne l$ , consideriamo una classe di determinanti di Vandermonde generalizzati di ordine quattro $G\left(a;x\right)$, al variare di $x$ in ${\mathbb{Z}}^4$ , connessa con alcuni problemi diofantei. Dimostriamo che il numero delle soluzioni $y\in {\mathbb{Z}}^3$ in posizione generica dell'equazione polinomiale-esponenziale disomogenea $G\left(a;0,y\right)=0$ non supera una costante esplicita $N\left(d\right)$ dipendente solo da $d=\left[\mathbb{Q}\right(\alpha {}_1,\mathrm{\dots },\alpha {}_m):\mathbb{Q}]$.

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:(i)Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?(ii)Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?

We study the Diophantine equations ${(k!)}^n-k^n={(n!)}^k-n^k$ and ${(k!)}^n+k^n={(n!)}^k+n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.

The ring of power sums is formed by complex functions on $\mathbb{N}$ of the form$$\alpha \left(n\right)=b_1{c}_1^n+b_2{c}_2^n+...+b_h{c}_h^n,$$for some $b_i\in \overline{\mathbb{Q}}$ and $c_i\in \mathbb{Z}$. Let $F(x,y)\in \overline{\mathbb{Q}}[x,y]$ be absolutely irreducible, monic and of degree at least $2$ in $y$. We consider Diophantine inequalities of the form$$\left|F(\alpha \left(n\right),y)\right|\<|\frac{\partial F}{\partial y}(\alpha \left(n\right),y)|·{\left|\alpha \left(n\right)\right|}^{-\epsilon }$$and show that all the solutions $(n,y)\in \mathbb{N}\times \mathbb{Z}$ have $y$ parametrized by some power sums in a finite set. As a consequence, we prove that the equation$$F\left(\alpha \right(n),y)=f\left(n\right),$$with $f\in \mathbb{Z}\left[x\right]$ not constant, $F$ monic in $y$ and $\alpha$ not constant, has only finitely many solutions.

In this paper, we study triples $a,b$ and $c$ of distinct positive integers such that $ab+1,ac+1$ and $bc+1$ are all three members of the same binary recurrence sequence.

In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.Moreover, we study similar problems in this context as the equation $G_n\left(x\right)=G_m\left(P\left(x\right)\right),(m,n)\in {\mathbb{N}}^2$, where $\left(G_n\left(x\right)\right)$ is a linear recurring sequence of polynomials and $P\left(x\right)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations $f\left(x\right)=\delta y^m$ in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and $m\in {\mathbb{Z}}_{\ge 2}$. Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F, f, m. These...