title: Lucas chain
text: # Lucas chain
  
  In mathematics, a Lucas chain is a restricted type of addition chain, named for the French mathematician Édouard Lucas. It is a sequence
  
      $a_0,$ $a_1,$ $a_2,$ $a_3,$ ...
  
  that satisfies
  
      $a_0=1,$
  
  and
  
      for each k > 0: $a_k$ = $a_i$ + $a_j,$ and either $a_i$ = $a_j$ 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 n, then Kutz has shown that most n do not have L < (1-ε) $log_φ$ n, where φ is the Golden ratio.
  
  