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Sequences and series

In mathematics, an addition chain is a sequence a₀, a₁, a₂, a₃, ... that satisfies

a₀ = 1, and

for each k>0: a_k = a_i + a_j for some i, j < k.

As an example: 1, 2, 3, 6, 12, 24, 30, 31 is an addition chain for 31, of length 7, since

2 = 1 + 1

3 = 2 + 1

6 = 3 + 3

12 = 6 + 6

24 = 12 + 12

30 = 24 + 6

31 = 30 + 1

Addition chains can be used for exponentiation: so for example we only need 7 multiplications to calculate 5³¹:

5² = 5¹ × 5¹

5³ = 5² × 5¹

5⁶ = 5³ × 5³

5¹² = 5⁶ × 5⁶

5²⁴ = 5¹² × 5¹²

5³⁰ = 5²⁴ × 5⁶

5³¹ = 5³⁰ × 5¹

External links

http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html

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Topics: Addition Chain, Addition In N, Additional Member System...

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