Using Lucas Sequences to Generalize a Theorem of Sierpiński

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In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k· 2ⁿ+1 is composite for all positive integers n. In this paper, we prove some generalizations of Sierpiński's theorem with 2ⁿ replaced by expressions involving certain Lucas sequences Uₙ(α,β). In particular, we show the existence of infinitely many Lucas pairs (α,β), for which there exist infinitely many positive integers k, such that k (Uₙ(α,β)+(α-β)²)+1 is composite for all integers n≥ 1. Sierpiński's theorem is the special case of α=2 and β=1. Finally, we establish a nonlinear version of this result by showing that there exist infinitely many rational integers α>1, for which there exist infinitely many positive integers k, such that k² (Uₙ(α,1)+(α-1)²)+1 is composite for all integers n≥ 1.