EXPLICIT BOUNDS FOR THE DIOPHANTINE EQUATION A!B! = C!
Résumé
A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A ≤ B ≤ C − 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6. Several estimates on the relative size of the parameters are known, such as the one given by Erdös C − B ≤ 5 log log C, or the one given by Bhat and Ramachandra C −B ≤ (1/ log 2+o(1)) log log C. We check the conjecture for B ≤ 10 3000 and give better explicit bounds such as C − B ≤ log log(B+1) log 2 − 0.8803.
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)
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