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Uniform lower bound for the least common multiple of a polynomial sequence. (Une borne inférieure uniforme pour le plus petit commun multiple d’une suite polynomiale.) (English. French summary) Zbl 1280.11064

Summary: Let \(n\) be a positive integer and \(f(x)\) be a polynomial with nonnegative integer coefficients. The authors prove that \(\mathrm{lcm}_{\lceil n/2 \rceil\leq i\leq n}\{f(i)\}\geq 2^n\), except that \(f(x)=x\) and \(n=1,2,3,4,6\) and that \(f(x)=x^s\), with \(s\geq 2\) being an integer and \(n=1\), where \(\lceil n/2 \rceil\) denotes the smallest integer, which is not less than \(n/2\). This improves and extends the lower bounds obtained by M. Nair [Am. Math. Mon. 89, 126–129 (1982; Zbl 0494.10004)], B. Farhi [J. Number Theory 125, No. 2, 393–411 (2007; Zbl 1124.11005)] and S. M. Oon [Abstr. Appl. Anal. 2013, Article ID 218125, 4 p. (2013; Zbl 1267.11004)]

MSC:

11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11B83 Special sequences and polynomials
11C08 Polynomials in number theory
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References:

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