Let (Fn)n≥1 be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that log lcm(F1,F2,...,Fn) ∼ 3log α π2 ⋅ n2,as n → +∞, where lcm is the least common multiple and α:= 1 + 5)/2 is the golden ratio. We prove that for every periodic sequence s = (sn)n≥1 in {-1, +1} there exists an effectively computable rational number Cs > 0 such that log lcm(F3 + s3,F4 + s4,...,Fn + sn) ∼ 3log α π2 ⋅ Cs ⋅ n2,as n → +∞. Moreover, we show that if (sn)n≥1 is a sequence of independent uniformly distributed random variables in {-1, +1} then [log lcm(F3 + s3,F4 + s4,...,Fn + sn)] ∼ 3log α π2 ⋅15Li2(1/16) 2 ⋅ n2,as n → +∞, where Li2 is the dilogarithm function.

On the l.c.m. of shifted Fibonacci numbers / Sanna, Carlo. - In: INTERNATIONAL JOURNAL OF NUMBER THEORY. - ISSN 1793-0421. - STAMPA. - 17:9(2021), pp. 2009-2018. [10.1142/S1793042121500743]

On the l.c.m. of shifted Fibonacci numbers

Sanna Carlo
2021

Abstract

Let (Fn)n≥1 be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that log lcm(F1,F2,...,Fn) ∼ 3log α π2 ⋅ n2,as n → +∞, where lcm is the least common multiple and α:= 1 + 5)/2 is the golden ratio. We prove that for every periodic sequence s = (sn)n≥1 in {-1, +1} there exists an effectively computable rational number Cs > 0 such that log lcm(F3 + s3,F4 + s4,...,Fn + sn) ∼ 3log α π2 ⋅ Cs ⋅ n2,as n → +∞. Moreover, we show that if (sn)n≥1 is a sequence of independent uniformly distributed random variables in {-1, +1} then [log lcm(F3 + s3,F4 + s4,...,Fn + sn)] ∼ 3log α π2 ⋅15Li2(1/16) 2 ⋅ n2,as n → +∞, where Li2 is the dilogarithm function.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11583/2898894