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The Scholz-Brauer problem on addition chains.

An addition chain for a positive integer n is a set 1 = «Ό < CLi < < a r = n of integers such that every element α* is the sum aj + a k of two preceding members (not necessarily distinct) of the set. The smallest length r for which an addition chain for n exists is denoted by l(n). Let λ(n) = [Iog2^], and let v(n) denote the number of ones in the binary representation of n. The purpose of this paper is to show how to establish the result that if v(n) ^ 9 then l(ri) Ξ> λ(n) + 4. This is the m = S case of the conjecture that if v{n) ^ 2 m -f-1 then l(n) Ξ> λ(n) 4- m + 1 for which cases m = 0,1, 2 have previously been estabished. The fact that the conjecture is true for m = 3 leads to the theorem that n = 2 m (23) + 7 for m ^ 5 is an infinite class of integers for which l(2ri) = l(n). The paper concludes with this result.