Fibonacci Variations of a Conjecture of Polignac
نویسنده
چکیده
In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form 2n + p, for some integer n ≥ 0 and some prime p. In 1950, Erdős constructed infinitely many counterexamples to Polignac’s conjecture. In this article, we show that there exist infinitely many positive integers that cannot be written in either of the forms Fn + p or Fn− p, where Fn is a Fibonacci number, and p is a prime.
منابع مشابه
The lower bound for the number of 1-factors in generalized Petersen graphs
In this paper, we investigate the number of 1-factors of a generalized Petersen graph $P(N,k)$ and get a lower bound for the number of 1-factors of $P(N,k)$ as $k$ is odd, which shows that the number of 1-factors of $P(N,k)$ is exponential in this case and confirms a conjecture due to Lovász and Plummer (Ann. New York Acad. Sci. 576(2006), no. 1, 389-398).
متن کاملA Study of Relationship Among Goldbach Conjecture, Twin Prime and Fibonacci Number
In 2015, Liu et al. proposed a study relationship between RSA public key cryptosystem and Goldbach’s conjecture properties. They discussed the relationship between RSA and Goldbach conjecture, twin prime and Goldbach conjecture. In this paper the author will extend to introduce the relationsip among Goldbach conjecture, twin prime and Fibonacci number. Based on their contribution, the author co...
متن کاملFibonacci-like behavior of the number of numerical semigroups of a given genus
We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the number of semigroups of genus at most 50. The Wilf conjecture has also been checked for all numerical semigroups with genus in the same range.
متن کاملEnergy 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.
متن کاملOn Fibonacci Powers
Fibonacci numbers have engaged the attention of mathematicians for several centuries, and whilst many of their properties are easy to establish by very simple methods, there are several unsolved problems connected to them. In this paper we review the history of the conjecture that the only perfect powers in Fibonacci sequence are 1, 8, and 144. Afterwards we consider more stronger conjecture an...
متن کامل