Let (Fn)n≥1 be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that loglcm(F1,F2,…,Fn) ∼ 3logα π2 ⋅ n2,asn → +∞, 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 loglcm(F3 + s3,F4 + s4,…,Fn + sn) ∼ 3logα π2 ⋅ Cs ⋅ n2,asn → +∞. Moreover, we show that if (sn)n≥1 is a sequence of independent uniformly distributed random variables in {−1, +1} then 𝔼[loglcm(F3 + s3,F4 + s4,…,Fn + sn)] ∼ 3logα π2 ⋅15Li2(1/16) 2 ⋅ n2,asn → +∞, where Li2 is the dilogarithm function.

中文翻译：

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在移动的斐波那契数的 lcm 上

让(Fn)n≥1是斐波那契数列。Guy 和 Matiyasevich 证明了 日志厘米(F1,F2,…,Fn) ～ 3日志α π2 ⋅ n2,作为n → +∞, 其中 lcm 是最小公倍数并且α ：= 1 + 5)/2是黄金比例。我们证明对于每个周期序列s = (sn)n≥1在{-1, +1}存在一个有效可计算的有理数Cs > 0这样 日志厘米(F3 + s3,F4 + s4,…,Fn + sn) ～ 3日志α π2 ⋅ Cs ⋅ n2,作为n → +∞. 此外，我们证明如果(sn)n≥1是一系列独立的均匀分布的随机变量{-1, +1}然后 𝔼[日志厘米(F3 + s3,F4 + s4,…,Fn + sn)] ～ 3日志α π2 ⋅15李2(1/16) 2 ⋅ n2,作为n → +∞, 在哪里李2是对数函数。