In mathematics, a Lucas chain is a restricted type of addition chain, named for the French mathematician ÃÂdouard Lucas. It is a sequence
that satisfies , and, for each ,
and either
for some .
The sequence of powers of 2 (1, 2, 4, 8, 16, ...) and the Fibonacci sequence (with a slight adjustment of the starting point 1, 2, 3, 5, 8, ...) are simple examples of Lucas chains.
Lucas chains were introduced by Peter Montgomery in 1983. If is the length of the shortest Lucas chain for , then Kutz has shown that most do not have , where ÃÂ is the Golden ratio.