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Density of solutions to quadratic congruences

Neha Prabhu (2017)

Czechoslovak Mathematical Journal

A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k > 1 . Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n ≤ x with k prime factors such that a fixed quadratic equation has exactly 2 k solutions modulo n .

Determinanti polinomiali-esponenziali

Raffaele Marcovecchio (2004)

Bollettino dell'Unione Matematica Italiana

Dati m = 2 o m = 3 numeri algebrici non nulli α = α 1 , … , α m tali che α j / α l non è una radice dell'unità per ogni j ≠ l , consideriamo una classe di determinanti di Vandermonde generalizzati di ordine quattro G ⁢ a ; x , al variare di x in Z 4 , connessa con alcuni problemi diofantei. Dimostriamo che il numero delle soluzioni y ∈ Z 3 in posizione generica dell'equazione polinomiale-esponenziale disomogenea G ⁢ a ; 0 , y = 0 non supera una costante esplicita N ⁢ d dipendente solo da d = [ Q ( α 1 , … , α m ) : Q ] .

Diophantine equations after Fermat’s last theorem

Samir Siksek (2009)

Journal de Théorie des Nombres de Bordeaux

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, ...?

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