Addition chain

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.

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 2 ⁡ ( n ) + log 2 ⁡ ( ν ( n ) ) − 2.13 ≤ l ( n ) ≤ log 2 ⁡ ( n ) + log 2 ⁡ ( n ) ( 1 + o ( 1 ) ) / log 2 ⁡ ( log 2 ⁡ ( n ) ) ,

where ν ( n ) is Hamming weight (the number of ones) of the 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.

The first integer with l(2n) < l(n) is n = 375494703, which is followed by 602641031, 619418303, and so on (sequence A230528 in the OEIS).

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

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) .

It is known to be true for Hansen numbers, a generalization of Brauer numbers; N. Clift checked by computer that all n≤5784688 are Hansen (while 5784689 is not). Clift further checked that is true with equality for n≤64.