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B. Brindza and A. Pint´ ´er (Debrecen)To the memory of Paul Erd˝osLet f(X) and g(Y ) be polynomials with integral coecients in the singleindependent variables X and Y . The diophantine problem f(x) = g(y) isstrongly related to the absolute irreducibility and the genus of f(X)−g(Y )as pointed out by Davenport, Lewis and Schinzel [DLS]:Theorem A. Let f(X) be of degree n > 1 and g(Y ) of degreem > 1. Let D(λ) = disc(f(x) + λ) and E(λ) = disc(g(y) + λ). Supposethere are at least [n/2] distinct roots of D(λ) = 0 for which E(λ) 6= 0 . Thenf(X)−g(Y ) is irreducible over the complex eld. Further, the genus of theequation f(x) − g(y) = 0 is strictly positive except possibly when m = 2 orm = n = 3. Apart from these possible exceptions, the equation has at mosta nite number of integral solutions.The purpose of this note is to handle some special cases. For an integerk > 1 we setf

On the irreducibility of some polynomials in two variables