An improved bound on the least common multiple of polynomial sequences
Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 891-899.

Cilleruelo conjectured that if f[x] of degree d2 is irreducible over the rationals, then loglcm(f(1),...,f(N))(d-1)NlogN as N. He proved it for the case d=2. Very recently, Maynard and Rudnick proved there exists c d >0 with loglcm(f(1),...,f(N))c d NlogN, and showed one can take c d =d-1 d 2 . We give an alternative proof of this result with the improved constant c d =1. We additionally prove the bound logradlcm(f(1),...,f(N))2 dNlogN and make the stronger conjecture that logradlcm(f(1),...,f(N))(d-1)NlogN as N.

Cilleruelo a conjecturé que si f[x] de degré d2 est irréductible sur les rationnels, alors loglcm(f(1),...,f(N))(d-1)NlogN quand N. Il l’a prouvé dans le cas d=2. Très récemment, Maynard et Rudnick ont prouvé qu’il existe c d >0 tel que loglcm(f(1),...,f(N))c d NlogN, et ont montré qu’on peut prendre c d =d-1 d 2 . Nous donnons une preuve alternative de ce résultat avec la constante améliorée c d =1. De plus, nous prouvons la minoration logradlcm(f(1),...,f(N))2 dNlogN et proposons une conjecture plus forte affirmant que logradlcm(f(1),...,f(N))(d-1)NlogN quand N.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1146
Classification: 11N32
Keywords: Least common multiple, polynomial sequence
Sah, Ashwin 1

1 Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA
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Sah, Ashwin. An improved bound on the least common multiple of polynomial sequences. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 891-899. doi : 10.5802/jtnb.1146. http://archive.numdam.org/articles/10.5802/jtnb.1146/

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