Lucas chain

Template:Short description In mathematics, a Lucas chain is a restricted type of addition chain, named for the French mathematician Édouard Lucas. It is a sequence

$a_0,a_1,a_2,a_3,\dots$

that satisfies a₀=1, and, for each k > 0,

$a_k=a_i+a_j,$

and either

$a_i=a_j\text{ or }|a_i-a_j|=a_m$

for some i, j, m < k.

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 L(n) is the length of the shortest Lucas chain for Template:Mvar, then Kutz has shown that most Template:Mvar do not have L < (1-ε) log_φ(n), where φ is the Golden ratio.