On the Finiteness of Certain Rabinowitsch Polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x2+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for t=[<f>√ of m</f>] and consecutive integers x=x0, x0+1, …, x0+t−1, &z.sfnc;f(x)&z.sfnc; is either 1 or prime. In this note, we show that there are only finitely many Rabinowitsch polynomials fm(x) such that 1+4m is square free. [Copyright &y& Elsevier]