On the Integrity of Certain Fibonacci Sums
نویسندگان
چکیده
[n is an arbitrary natural number, r is an arbitrary (nonzero) real quantity) gives & positive integer k. Since both r and k turn out to be Fibonacci number ratios, the results established in this paper can be viewed as a particular kind of Fibonacci identities that are believed to be new [see (4.7) and (4.8)]. Throughout the paper we shall make use of the following properties of the Fibonacci numbers and of the Lucas numbers Ln which are either available in [5] and [11] or can be readily derived by using the Binet forms for Fn and Ln: F2n=FnLn, (1.4) 5i?=/*-4(-l)\ (1.5) L2„-2(-l)" = 5Fn, (1.6) F„ divides Fk iff n divides k (for n>3), (1.7) L„ = Lk (mod 5) iff n = * (mod 4), (1.8) L„+k-(-l)Ln_k=5F„Fk, (1.9) A»*+(-i)*4-* = 4 4 0-io)
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