## 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

### Keywords:

uniform lower bound; polynomial sequence

### Citations:

Zbl 0494.10004; Zbl 1267.11004; Zbl 1124.11005
Full Text:

### References:

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