The scholz-brauer problem on addition chains

Abstract

An addition chain for a positive integer n is a set 1 = α0 < α1 < αr = n of integers such that every element di is the sum αj + αkof two preceding members (not neces- sarily distinct) of the set. The smallest length r for which an addition chain for n exists is denoted by l(n). Let λ(n) = [log2n], 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(n) ≧λ(n) + 4. This is the m = 3 case of the conjecture that if v(n) ≧2m + 1 then l(n) ≧λ(n) + 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 = 2m(23) + 7 for m ≧5 is an infinite dass of integers for which l(2n) = l(n). The paper concludes with this result. © 1973 Pacific Journal of Mathematics.