There has been interest during the last decade in properties of the sequence , , where are fixed (multiplicatively independent) elements in one of , or . In the case of , Bugeaud, Corvaja and Zannier have obtained an upper bound for any given and all large , and demonstrate its sharpness by extracting from a paper of Adleman, Pomerance, and Rumely a lower bound for infinitely many , where is an absolute constant. Silverman has proved an analogous lower bound for infinitely many , over . This paper generalizes Silverman’s theorem to for any positive integer , where is the th cyclotomic polynomial, Silverman’s result being the case . Over , the lower bound has been proved in the first author’s Ph.D. thesis for the case , i.e. for . Here we prove a conditional result that the lower bound for arbitrary holds over under GRH (the generalized Riemann Hypothesis).
Les propriétés des suites , , où sont des éléments fixés (multiplicativement indépendants) dans ou , ont été étudiées depuis des décennies. Dans le cas de , Bugeaud, Corvaja et Zannier ont obtenu une borne supérieure pour tout donné et tout grand, et montrent que la borne est optimale en extrayant la borne inférieure , pour une infinité de (où est une constante absolue), d’un article d’Adleman, Pomerance, et Rumely. Silverman a montré une borne inférieure analogue pour une infinité de , pour l’anneau . Ce travail généralise le théorème de Silverman à pour tout entier positif , où est le ième polynôme cyclotomique, le résultat de Silverman correspondant au cas . Sur , la borne inférieure a été montrée dans la thèse du premier auteur dans le cas , i.e. pour la suite . Ici nous montrons que la borne inférieure est valide sur pour tout , sous GRH.
Accepted:
Published online:
DOI: 10.5802/jtnb.893
Classification: 11A05, 11R47, 11N37
Keywords: Greatest common divisor, sequence, cyclotomic polynomial
@article{JTNB_2015__27_1_53_0, author = {Joseph Cohen and Jack Sonn}, title = {A cyclotomic generalization of the sequence $\gcd (a^n-1,b^n-1)$}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {53--65}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {27}, number = {1}, year = {2015}, doi = {10.5802/jtnb.893}, mrnumber = {3346964}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.893/} }
TY - JOUR TI - A cyclotomic generalization of the sequence $\gcd (a^n-1,b^n-1)$ JO - Journal de Théorie des Nombres de Bordeaux PY - 2015 DA - 2015/// SP - 53 EP - 65 VL - 27 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.893/ UR - https://www.ams.org/mathscinet-getitem?mr=3346964 UR - https://doi.org/10.5802/jtnb.893 DO - 10.5802/jtnb.893 LA - en ID - JTNB_2015__27_1_53_0 ER -
Joseph Cohen; Jack Sonn. A cyclotomic generalization of the sequence $\gcd (a^n-1,b^n-1)$. Journal de Théorie des Nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 53-65. doi : 10.5802/jtnb.893. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.893/
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