Products of Factorials in Smarandache Type Expressions

In [3] and [5] the authors ask how many primes are of the form xY + yX, where gcd (x, y) = 1 and x, y 2: 2. Moreover, Jose Castillo (see [2]) asks how many primes are of the Smarandache form xil + X2 X3 + ... + Xn Xl , where n > 1, Xl, X2, ••• , Xn > 1 and gcd (Xl, X2, ••• , X n ) = 1 (see [9]). In this article we announce a lower bound for the size of the largest prime divisor of an expression of the type axY + byx, where ab f= 0, x, y 2: 2 and gcd (x, y) = l. For any finite extension F of Q let dF = [F : Q]. For any algebraic number ( E F let NF (() denote the norm of (. For any rational integer n let Pen) be the largest prime number P diViding n with the convention that P(O) = P(±l) = l.