Cilleruelo conjectured that if of degree is irreducible over the rationals, then as . He proved it for the case . Very recently, Maynard and Rudnick proved there exists with , and showed one can take . We give an alternative proof of this result with the improved constant . We additionally prove the bound and make the stronger conjecture that as .
Cilleruelo a conjecturé que si de degré est irréductible sur les rationnels, alors quand . Il l’a prouvé dans le cas . Très récemment, Maynard et Rudnick ont prouvé qu’il existe tel que , et ont montré qu’on peut prendre . Nous donnons une preuve alternative de ce résultat avec la constante améliorée . De plus, nous prouvons la minoration et proposons une conjecture plus forte affirmant que quand .
Revised:
Accepted:
Published online:
Classification: 11N32
Keywords: Least common multiple, polynomial sequence
Author's affiliations:
@article{JTNB_2020__32_3_891_0, author = {Ashwin Sah}, title = {An improved bound on the least common multiple of polynomial sequences}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {891--899}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {32}, number = {3}, year = {2020}, doi = {10.5802/jtnb.1146}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1146/} }
TY - JOUR TI - An improved bound on the least common multiple of polynomial sequences JO - Journal de Théorie des Nombres de Bordeaux PY - 2020 DA - 2020/// SP - 891 EP - 899 VL - 32 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1146/ UR - https://doi.org/10.5802/jtnb.1146 DO - 10.5802/jtnb.1146 LA - en ID - JTNB_2020__32_3_891_0 ER -
Ashwin Sah. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1146/
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