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On the discriminator of Lucas sequences

We consider the family of Lucas sequences uniquely determined by \(U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),\) with initial values \(U_0(k)=0\) and \(U_1(k)=1\) and \(k\ge 1\) an arbitrary integer. For any integer \(n\ge 1\) the discriminator function \(\mathcal {D}_k(n)\) of \(U_n(k)\) is defined as the smallest integer m such that \(U_0(k),U_1(k),\ldots ,U_{n-1}(k)\) are pairwise...

Romanov type problems

Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form \(n=p+2^{2^k}+m!\) and \(n=p+2^{2^k}+2^q\) where \(m,k \in \mathbb {N}\) and p, q are primes. In the opposite direction, Erdős constructed a full arithmetic progression of odd integers none of...

Diophantine Triples and k-Generalized Fibonacci Sequences

We show that if \(k\ge 2\) is an integer and \(\big (F_n^{(k)}\big )_{n\ge 0}\) is the sequence of k-generalized Fibonacci numbers, then there are only finitely many triples of positive integers \(1<a<b<c\) such that \(ab+1,~ac+1,~bc+1\) are all members of \(\big \{F_n^{(k)}: n\ge 1\big \}\). This generalizes a previous result where the statement for \(k=3\) was proved. The...

On the Local Minima of the Order of Appearance Function

Received 17 August 2015; Accepted 20 September 2015 Academic Editor: Pentti Haukkanen Copyright © 2015 Florian Luca and Thato Mosima. This is an open access article distributed under the Creative Commons