A REMARK ON THE DISTRIBUTION OF ADDITION CHAINS THEOPHILUS AGAMA Abstract. We prove the prime obstruction principle and the sparsity law. These two are collective assertions that there cannot be many primes in an addition chain. 1. Introduction An addition chain of length h leading to n is a sequence of numbers s o =1,s 1 = 2,...,s h = n where s i = s k + s s for i>k ≥ s ≥ 0. The number of terms (excluding the first term) in an addition chain leading to n is the length of the chain. We call an addition chain leading to n with a minimal length an optimal addition chain leading to n. In standard practice, we denote by ι(n) the length of an optimal addition chain that leads to n. Example 1.1. The following is an example of an addition chain that leads to 15: 1, 2, 3, 5, 6, 8, 11, 14, 15 obtained from the sequence of additions 2=1+1, 3=2+1, 5=2+3, 6=3+3, 8=3+5, 11 = 5 + 6, 14 = 6 + 8, 15 = 14 + 1. We remark that the same addition chain can also be obtained from the sequence of additions 2 = 1+1, 3 = 2+1, 5 = 2+3, 6 = 5+1, 8 = 6+2, 11 = 8+3, 14 = 11+3, 15 = 14+1. The possibility to obtain an addition chain using distinct sequence of additions creates a subtle ambiguity. It suggests that knowing an addition chain leading to a fixed positive integer without specifying how the terms were obtained may be unsatisfactory, as it hides the procedure for obtaining the terms in the chain. The underlying intrinsic lack of uniqueness for this construction may be resolved by rewriting each term in the chain as the sum using the immediately previous term in the chain. However, an addition chain may not necessarily use the immediately previous term to generate the next term in the chain, so that in this case at most a term in the sum may not be a previous term in the chain. Current research on addition chains focuses mainly on optimizing the length of an addition chain leading to a fixed positive integer n. Despite extensive work on the topic, there is no known asymptotic formula for the optimal length of an addition chain that leads to fixed positive integers n. To that effect, improving the upper Date : May 10, 2025. 2010 Mathematics Subject Classification. Primary 11Pxx, 11Bxx; Secondary 11Axx, 11Gxx. Key words and phrases. prime obstruction principle, sparsity law. 1