# Search: authors:"Florian Luca"

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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