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: ak = ak-1 + aj 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.
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Famous quotes containing the word chain:
“How have I been able to live so long outside Nature without identifying myself with it? Everything lives, moves, everything corresponds; the magnetic rays, emanating either from myself or from others, cross the limitless chain of created things unimpeded; it is a transparent network that covers the world, and its slender threads communicate themselves by degrees to the planets and stars. Captive now upon earth, I commune with the chorus of the stars who share in my joys and sorrows.”
—Gérard De Nerval (1808–1855)