Abstract
Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + ℓ for | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.
Highlights
The study of Diophantine equations was started by Diophantus of Alexandria in around the third century BC
For example, the proof of the unsolvability of Fermat’s equation in 1995 and compare it with a much simpler matter to prove that similar equation x n + yn = zn+1 has an infinite family of solutions given by xk = kn + 1, yk = k(kn + 1) and zk = kn + 1, for all k ∈ N and n ≥ 2
Fibonacci sequence ( Fn )n≥0, which corresponds to the choice of a = 0 and b = 1, is surely the most famous from these sequences (the well-known Lucas sequence ( Ln )n≥0 is created by choice a = 2 and b = 1)
Summary
Another popular area of research that has drawn great interest is the study of the divisibility properties of Fibonacci numbers It is still an open problem if there are infinitely many primes in the Fibonacci sequence (we recommend [2,3,4,5,6]). All solutions of equations z(n) = n ± 1 are prime numbers This fact was proven by Lehmer in Theorem 5.1 in [27]. A new question arises: what are the solutions of Diophantine equation z(n) = P(n), for a given polynomial P( x ) ∈ Z[ x ] with positive coefficients? To prove that all solutions of equation z(n) = z(n2 ) are prime numbers is implied in the first case of Fermat’s Last Theorem (see [28]).
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