Fibonacci Variations of a Conjecture of Polignac

In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form , for some integer and some prime . In 1950, Erdős constructed infinitely many counterexamples to Polignac's conjecture. In this article, we show that there exist infinitely many positive intege...

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Veröffentlicht in:	Integers (Berlin, Germany) Germany), 2012-08, Vol.12 (4), p.659-667
1. Verfasser:	Jones, Lenny
Format:	Artikel
Sprache:	eng
Schlagworte:	Fibonacci Number Polignac Prime Number
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Beschreibung
Zusammenfassung:	In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form , for some integer and some prime . 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 or , where is a Fibonacci number, and is a prime.
ISSN:	1867-0652 1867-0652 1867-0660
DOI:	10.1515/integers-2011-0126