The "factor method" for finding addition chains is based on the prime factorization of the number formula_1 to be represented. If formula_1 has a number formula_9 as one of its prime factors, then an addition chain for formula_1 can be obtained by starting with a chain for formula_11, and then concatenating onto it a chain for formula_9, modified by multiplying each of its numbers by formula_11. The ideas of the factor method and binary method can be combined into "Brauer's m-ary method" by choosing any number formula_14 (regardless of whether it divides formula_1), recursively constructing a chain for formula_16, concatenating a chain for formula_14 (modified in the same way as above) to obtain formula_18, and then adding the remainder. Additional refinements of these ideas lead to a family of methods called "sliding window methods".