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Fibonacci Variations of a Conjecture of Polignac

Abstract. In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form , for some integer 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 or , where is a Fibonacci number, and p is a prime.