No abstract

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) isso thatUsing this relation, Churchhouse has shown thatandwherewithMoreoverand b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,andThe object of this note is to prove these conjectures.

No abstract

A histori cal survey of some aspects of the theory of partitions is given here.