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)]


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


[1] Alaca, S.; Williams, K. S., Introductory Algebraic Number Theory (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1035.11001
[2] Bateman, P.; Kalb, J.; Stenger, A., A limit involving least common multiples, Amer. Math. Monthly, 109, 393-394 (2002) · Zbl 1124.11300
[3] Chebyshev, P. L., Memoire sur les nombres premiers, J. Math. Pures Appl., 17, 366-390 (1852)
[4] Farhi, B., Minoration non triviales du plus petit commun multiple de certaines suites finies dʼentiers, C. R. Acad. Sci. Paris, Ser. I, 341, 469-474 (2005) · Zbl 1117.11005
[5] Farhi, B., Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory, 125, 393-411 (2007) · Zbl 1124.11005
[6] Farhi, B.; Kane, D., New results on the least common multiple of consecutive integers, Proc. Amer. Math. Soc., 137, 1933-1939 (2009) · Zbl 1229.11007
[7] Hanson, D., On the product of the primes, Canad. Math. Bull., 15, 33-37 (1972) · Zbl 0231.10008
[8] Hong, S.; Feng, W., Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Acad. Sci. Paris, Ser. I, 343, 695-698 (2006) · Zbl 1156.11004
[9] Hong, S.; Kominers, S. D., Further improvements of lower bounds for the least common multiple of arithmetic progressions, Proc. Amer. Math. Soc., 138, 809-813 (2010) · Zbl 1196.11007
[10] Hong, S.; Qian, G., The least common multiple of consecutive arithmetic progression terms, Proc. Edinb. Math. Soc., 54, 431-441 (2011) · Zbl 1304.11008
[11] Hong, S.; Qian, G.; Tan, Q., The least common multiple of a sequence of products of linear polynomials, Acta Math. Hung., 135, 160-167 (2012) · Zbl 1265.11093
[12] Nair, M., On Chebyshev-type inequalities for primes, Amer. Math. Monthly, 89, 126-129 (1982) · Zbl 0494.10004
[13] Oon, S. M., Note on the lower bound of least common multiple, Abstr. Appl. Anal. (2013), Art. ID 218125, 4 p · Zbl 1267.11004
[14] Qian, G.; Hong, S., Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms, Arch. Math., 100, 337-345 (2013) · Zbl 1290.11014
[15] Wu, R.; Tan, Q.; Hong, S., New lower bounds for the least common multiples of arithmetic progressions, Chin. Ann. Math., Ser. B (2013) · Zbl 1292.11006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.