| 释义 |
Lucas Math.
(ˈl(j)uːkəs, ‖ lyka)
The name of F. Édouard A. Lucas (1842–91), French mathematician, used attrib. to designate (a) the sequence of integers 1, 3, 4, 7,{ddd}, formed in the same way as the Fibonacci numbers; (b) the sequences generated by the recurrence relation un + 2 = Pun+1- Qun when u0 = 0, u1 = 1 (the Fibonacci numbers being a particular case corresponding to P = 1, Q = -1) and when u0 = 2, u1 = P, which are respectively defined by un = (an - bn)/(a - b) and un = an + bn(n = 0, 1, 2,..), where a and b are the roots of x2 - Px + Q = 0.
[1919L. E. Dickson Hist. Theory Numbers I. xvii. 393 (heading) Recurring series; Lucas' un, vn.] 1953Scripta Math. XIX. 278 (heading) Linear expressions for the powers of Fibonacci and Lucas numbers. Ibid., The ith term of the Lucas sequence 1, 3, 4, 7... 1954Duke Math. Jrnl. XXI. 608 If any term of (W) vanishes, (W) is essentially the well-known Lucas sequence Ln = (αn - βn)/(α- β). 1961Pacific Jrnl. Math XI. 385 It would be interesting to make a numerical study of several recurrences..to endeavor to find out whether the two Lucas sequences 0, 1, P, {ddd} and 2, P, P2 - 2Q, {ddd} and their translates are essentially the only ones for which a global characterization of the divisors is possible. 1966Ogilvy & Anderson Excursions in Number Theory 164 The Lucas numbers satisfy the same recursion relation as the Fibonacci numbers, but have starting values L1 = 1, L2 = 3. 1972P. Ribenboim Algebraic Numbers i. 8 Prove: (a) Two consecutive Lucas numbers are relatively prime. (b) bn2 - bn - 1bn + 1 = (- 1)n.5 for every n≥ 1. |