On the Number of Minimal Addition Chains

An addition chain for a natural number n is a sequence 1 = a0 < a1 < . . . < ar = n of numbers such that for each 0 < i r, ai = aj + ak for some 0 k j < i. Thurber [9] introduced the function NMC(n) which denotes the number of minimal addition chains for a number n. Thurber calculated NMC(n) for some classes of n, such as when n has one or two ones in its binary representation. Also, he calculated NMC(2m n), for n X = {2, 3, 5, 9, 15, 17, 33, 49, 51, 65, 85, 97, 99}. For odd n does not belong to X and less than or equal to 127, he conjectured formula for NMC(2m n). In this paper, we verified the conjectures computationally up to m = 150 for each n. For n = 69, 75, and 109 the minimum value of m is corrected. For n = 57, and 111, the formula for NMC is corrected (there is a mistyping).

• Type: Conference
• Year: 2006
• Where: CSC
• Authors: Hatem M. Bahig
• Keywords: Addition Chains, CSC 2006, CSC 2007, Minimal Addition Chains, Natural Number