lt has been shown by different
methods that there is an infinity of positive odd integers not representable as the sum
of a prime and a (positive) power of 2, thus disproving a conjecture to the contrary
which had been made in the nineteenth century. The question then arises as to
whether or not all sufficiently large positive odd integers can be represented as the
sum of a prime and of two positive powers of 2; that is, as p + 2α+ 2b, where a,b > 0
and p is prime. (The corresponding question has been discussed for bases other
than 2 but is really quite trivial.) Theorem I gives a negative answer to this
question.
Theorem I. There is an infinity of distinct, positive odd integers not representable
as the sum of a prime and of two positive powers of 2.
