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 $d=\left[\mathbb{Q}\right(\alpha {}_1,\mathrm{\dots },\alpha {}_m):\mathbb{Q}]$.

In this paper the special diophantine equation $\frac{q^n-1}{q-1}=y$ with integer coefficients is discussed and integer solutions are sought. This equation is solved completely just for four prime divisors of $y-1$.

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, ...?

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and ${\mu }_i\in -1,1(i=1,2)$, and let $h(-2^{1-e}D)(e=0or1)$ denote the class number of the imaginary quadratic field $\mathbb{Q}\left(\surd (-2^{1-e}D)\right)$. In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then $h(-2^{1-e}D)\equiv 0\left(modn\right)$, where D, and n satisfy $kⁿ-2^{e+1}=Dx²$, x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.