Bulletin of the London Mathematical Society



Papers

ON A PROBLEM OF BROCARD


ALEXANDRU GICA a1 and LAURENTIU PANAITOPOL a1
a1 University of Bucharest, 14 Academiei St., RO-010014 Bucharest, Romania alex@al.math.unibuc.ro, pan@al.math.unibuc.ro

Article author query
gica a   [Google Scholar] 
panaitopol l   [Google Scholar] 
 

Abstract

It is proved that, if $P$ is a polynomial with integer coefficients, having degree 2, and $1>\varepsilon>0$, then $n(n-1)\cdots(n-k+1)=P(m)$ has only finitely many natural solutions $(m,n,k)$, $n\ge k>n\varepsilon$, provided that the $abc$ conjecture is assumed to hold under Szpiro's formulation.

(Received January 30 2004)
(Revised July 26 2004)

Maths Classification

11D75; 11J25; 11N13.