Lucas sequence has a great relationship with Fibonnacci sequence. So, after introducing Fibonacci numbers, I have to add another chapter on Lucas series. Firstly, the integers p and q are defined so that d = p2-4q >；; 0, thus the equation x2-Px+Q = 0 is obtained, and the roots of the equation are A and B. Now Lucas series is defined as Un(P, Q) = (an-bn)/(a-b) and Vn(P, Q) = an+bn, where n is a non-negative integer and U0(P, Q) = 0. Q) = 1, V0(P, Q) = 2, V 1(P, Q) = P, ... We have the following identities related to Lucas sequence: Um+n = UmVn-anbnUm-n, VM+n = vmvn-anbnmm. Vm- 1 (n = 1) U2n = UnVn，v2n = vn2-qnu2n+ 1 = UN+ 1vn-qn，v2n+ 1 = VN+ 1vn。 -1), we have Un as Fibonacci number, namely 0, 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89 and/kloc-. And Vn is Lucas number, that is, 2, 1, 3, 4, 7,1,18, 29, 47, 76, 123, 199 and 322. If (p, Q) = (2,-1), we have Un as the per number, that is, 0, 1, 2,5,12,29,70,169,408,985. And Vn is the Pell-Lucas number (see another article on Pell series), that is, 2, 2, 6, 14, 34, 82,198,478,154,2786. These are all famous sequences in mathematics. The properties of Lucas numbers Lucas numbers (Ln) have many properties similar to Fibonacci numbers. For example, Ln = Ln- 1+Ln-2, where the difference is L 1 = 1 and L2 = 3. So the Lucas numbers are: 1, 3, 4, 7, 1 1 8, 29, 47, 76, 123, ... (OEIS A 000204), where the square number is only. The prime numbers, namely Lucas prime numbers, are: 3,7, 1 1, 29,47. The largest possible prime number is known as L5742 19, with as many as 120005 digits. We have the following identities related to Lucas numbers: ln2-ln-1ln+1= 5 (-1) nl12+l22+...+ln2 = lnln+1-2lm+. 2 (where Fn is Fibonacci number) LM-N = (-1) N (LMLN-5 FMFN)/2LN2-5FN2 = 4 (-1) The year of digital discoverer of Lucas Prime Table N is 560031. Booker de Walter

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