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In mathematics, an addition chain for computing a positive integer n can be given by a sequence of natural numbers v and a sequence of index pairs w such that each term in v is the sum of two previous terms, the indices of those terms being specified by w:

v =(v₀,...,v_s), with v₀ = 1 and v_s = n

for each 0< i ≤ s holds: v_i = v_j + v_k, with w_i=(j,k) and 0 ≤ j,k ≤ i − 1

Often only v is given since it is easy to extract w from v, but sometimes w is not uniquely reconstructible. An introduction is given in.

Examples

As an example: v = (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 addition-chain 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

Methods for computing addition chains

Calculating an addition chain of minimal length is not easy; a generalized version of the problem, in which one must find a chain that simultaneously forms each of a sequence of values, is NP-complete. There is no known algorithm which can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or small memory usage. However, several techniques to calculate relatively short chains exist. One very well known technique to calculate relatively short addition chains is the binary method, similar to exponentiation by squaring. Other well-known methods are the factor method and window method.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Chain length

Let $l(n)$ denote the smallest s so that there exists an addition chain of length s which computes n. It is known that

$\log(n)+\log(\nu (n))-2.13\leq l(n)\leq \log(n)+\log(n)(1+o(1))/\log(\log(n))$,

where $\nu (n)$ is Hamming weight of binary expansion of n.

It is clear that l(2n) ≤ l(n)+1. Strict inequality is possible, as l(382) = l(191) = 11, observed by Knuth.

Brauer chain

A Brauer chain or star addition chain is an addition chain in which one of the summands is always the previous chain: that is,

for each k>0: a_k = a_k-1 + a_j for some j < k.

A Brauer number is one for which the Brauer chain is minimal.

Brauer proved that

l*(2ⁿ−1) ≤ n − 1 + l*(n)

where l* is the length of the shortest star chain. For many values of n,and in particular for n ≤ 2500, they are equal: l(n) = l*(n). But Hansen showed that there are some values of n for which l(n) ≠ l*(n), such as n = 2⁶¹⁰⁶ + 2³⁰⁴⁸ + 2²⁰³² + 2²⁰¹⁶ + 1 which has l*(n) = 6110, l(n) ≤ 6109.

Scholz conjecture

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The Scholz conjecture (sometimes called the Scholz–Brauer or Brauer–Scholz conjecture), named after A. Scholz and Alfred T. Brauer), is a conjecture from 1937 stating that

l(2ⁿ − 1) ≤ n − 1 + l(n) .

N. Clift checked this by computer for n ≤ 46. It is known to be true for Brauer numbers.

See also

• Addition chain exponentiation
• Addition-subtraction chain
• Vectorial addition chain
• Lucas chain

References

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External links

• http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html
• Template:SloanesRef
• F. Bergeron, J. Berstel. S. Brlek "Efficient computation of addition chains"