Perfect numbers and Fibonacci primes (I)

Fibonacci Numbers modulo Cubes of Primes

Let p be an odd prime. It is well known that Fp−( p 5 ) ≡ 0 (mod p), where {Fn}n>0 is the Fibonacci sequence and (−) is the Jacobi symbol. In this paper we show that if p 6= 5 then we may determine Fp−( p 5 ) mod p3 in the following way:

On perfect numbers which are ratios of two Fibonacci numbers ∗

Here, we prove that there is no perfect number of the form Fmn/Fm, where Fk is the kth Fibonacci number.

A Result about the Primes Dividing Fibonacci Numbers

Fibonacci Numbers at Most One Away from a Perfect Power

The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematic...

Energy of Graphs, Matroids and Fibonacci Numbers

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.