A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant Δn(a, b) of the trinomial fn,a,b(t) = tn + at + b, where n ≥ 5 is a fixed integer. In particular, it is shown that, under the abc-conjecture, for every n ≡ 1 (mod 4), the quadratic fields ( √ Δn(a, b) ) are pairwise distinct for a positive proportion of suchdiscriminants with integers a and b such that fn,a,b is irreducible over and |Δn(a, b)| ≤ X, as X →∞. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.
|Number of pages||8|
|Journal||Proceedings of the American Mathematical Society|
|Publication status||Published - Jan 2010|