Number Theory

Elementary Number Theory by W. Edwin Clark

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By W. Edwin Clark

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Now set n = 0, h = k. We derive from (20b) (22) It follows from (22) and (17c) that L 2h - 2( - l)h =SF~ (23) and from (23) and (17c) that (24) 5. We concentrate now on formula (24), a special case of the so-called Pell's equation. (Historians tell us that this is a misnomer, and the study of this type of Diophantine equation should more correctly be attributed to Fermat. ) (24) shows that F1i and Lh cannot have any common divisor larger than 2, and that either both Fh and L1i are even, or both are odd.

IV(b). n 2: G;+2G;-1 = G~+1 - Gf. (43) i= I 10. We have n n n n i= I i= I i= I i= 1 2: Gf= 2: G;(G;+ 1-G;_ 1)= 2: G;G;+ 1- 2: G;- 1 G; that is n 2: Gf = GnGn+1 -GoG1. i= 1 (44) Relationships 44 If G; [Ch. III =F;, then n L Ff= FnFn+l· (45) i= 1 This formula can also be illustrated geometrically. We do it in Fig. V(a). (See 13X21=F7 xF8 p2 7 p2 4 52 Fig. V(a). Ch. ) As n increases, the rectangle tends to a Gold Rectangle, because Jim n= F~+i tends oo n to 't (see (101) in Chapter VIII). In Fig. V(a) we have also drawn the straight lines which connect the centres C; of alternate squares.

1, hence a = 1, P= 1. Thus (57) p= -11\15. \15 and For the (58) (59) Formula (58) was known to De Moivre (1718) and rediscovered by Binet (1843), and also by Lame (1844) after whom the sequence is sometimes called. Fibonacci numbers as well as Lucas numbers are integers. tn and an are, of course, also generalized Fibonacci numbers, though integers only for n = 0. Nevertheless, formula (8) applies to them as well. For instance, with n = 0, From (58) and (59) it follows, trivially, that tn = l(Ln + \15Fn) and From (56) we notice that a and a= m n\15 2+10 an = i(L 11 - \15Fn).

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