Irreducible polynomials over F2r with three prescribed coefficients

For any positive integers n ≥ 3 and r ≥ 1, we prove that the number of monic irreducible polynomials of degree n over F2r in which the coefficients of Tn−1, Tn−2 and Tn−3 are prescribed has period 24 as a function of n, after a suitable normalization. A similar result holds over F5r, with the period being 60. We also show that this is a phenomena unique to characteristics 2 and 5. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements an equidistribution result of Katz