Gigantic pairs of minimal clones

L. Szabo (1992) asked for the minimal number n=n(|A|) such that the clone of all operations on A can be generated as the join of n minimal clones. He showed, e.g., n(p)=2 for any prime p, and later G. Czedli (1998) proved that if k has a divisor ⩾5 then n(k)=2. In this paper, a pair (f, g) of operations is called gigantic if each of f and g generates a minimal clone and the set {f, g} generates the clone of all operations. First, we give a general theorem to characterize a gigantic pair. Then we show that n(k)=2 for every k which is not a power of 2