An addition chain for a positive
integer n is a set 1 = a0< a1<⋯< ar= n of integers such that every element ai is
the sum aj+ ak 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) = [log2n], and let ν(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 ν(n) ≧ 9 then l(n) ≧ λ(n) + 4. This is the m = 3 case of the
conjecture that if ν(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 class of integers for which l(2n) = l(n). The paper concludes with this
result.
