A cyclotomic generalization of the sequence gcd(a n -1,b n -1)
Journal de Théorie des Nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 53-65.

Les propriétés des suites gcd(a n -1,b n -1), n=1,2,3,..., où a,b sont des éléments fixés (multiplicativement indépendants) dans ,[T] ou 𝔽 q [T], ont été étudiées depuis des décennies. Dans le cas de , Bugeaud, Corvaja et Zannier ont obtenu une borne supérieure exp(ϵn) pour tout ϵ>0 donné et tout n grand, et montrent que la borne est optimale en extrayant la borne inférieure exp(exp(clogn loglogn)), pour une infinité de n (où c est une constante absolue), d’un article d’Adleman, Pomerance, et Rumely. Silverman a montré une borne inférieure analogue deggcd(a n -1,b n -1)cn pour une infinité de n, pour l’anneau 𝔽 q [T]. Ce travail généralise le théorème de Silverman à gcd(Φ m (a n ),Φ m (b n )) pour tout entier positif m, où Φ m (x) est le mième polynôme cyclotomique, le résultat de Silverman correspondant au cas m=1. Sur , la borne inférieure a été montrée dans la thèse du premier auteur dans le cas m=2, i.e. pour la suite gcd(a n +1,b n +1). Ici nous montrons que la borne inférieure est valide sur pour tout m, sous GRH.

There has been interest during the last decade in properties of the sequence gcd(a n -1,b n -1), n=1,2,3,..., where a,b are fixed (multiplicatively independent) elements in one of ,[T], or 𝔽 q [T]. In the case of , Bugeaud, Corvaja and Zannier have obtained an upper bound exp(ϵn) for any given ϵ>0 and all large n, and demonstrate its sharpness by extracting from a paper of Adleman, Pomerance, and Rumely a lower bound exp(exp(clogn loglogn)) for infinitely many n, where c is an absolute constant. Silverman has proved an analogous lower bound deggcd(a n -1,b n -1)cn for infinitely many n, over 𝔽 q [T]. This paper generalizes Silverman’s theorem to gcd(Φ m (a n ),Φ m (b n )) for any positive integer m, where Φ m (x) is the mth cyclotomic polynomial, Silverman’s result being the case m=1. Over , the lower bound has been proved in the first author’s Ph.D. thesis for the case m=2, i.e. for gcd(a n +1,b n +1). Here we prove a conditional result that the lower bound for arbitrary m holds over under GRH (the generalized Riemann Hypothesis).

Reçu le : 2013-09-11
Accepté le : 2014-05-22
Publié le : 2015-05-21
DOI : https://doi.org/10.5802/jtnb.893
Classification : 11A05,  11R47,  11N37
Mots clés: 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},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {1},
     year = {2015},
     pages = {53-65},
     doi = {10.5802/jtnb.893},
     mrnumber = {3346964},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2015__27_1_53_0/}
}
Joseph Cohen; Jack Sonn. A cyclotomic generalization of the sequence  $\gcd (a^n-1,b^n-1)$. Journal de Théorie des Nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 53-65. doi : 10.5802/jtnb.893. https://jtnb.centre-mersenne.org/item/JTNB_2015__27_1_53_0/

[1] L.M. Adleman, C. Pomerance and R.S. Rumely, On distinguishing prime numbers from composite numbers, Ann. Math. 117, (1983), 173–206. | MR 683806 | Zbl 0526.10004

[2] N. Ailon and Z. Rudnick, Torsion points on curves and common divisors of a k -1,b k -1, Acta Arith. 113, (2004), 31–38. | MR 2046966 | Zbl 1057.11018

[3] Y. Bugeaud, P. Corvaja and U. Zannier, An upper bound for the G.C.D. of a n -1 and b n -1, Math. Zeit. 243, (2003), 79–84. | MR 1953049 | Zbl 1021.11001

[4] J. Cohen, Primitive roots in algebraic number fields, Ph.D. Thesis, Technion (2004).

[5] P. Corvaja and U. Zannier, A lower bound for the height of a rational function at S-unit points, Monatsh. Math. 144, 3 (2005), 203–224. | MR 2130274 | Zbl 1086.11035

[6] P. Corvaja and U. Zannier, Some cases of Vojta’s conjecture on integral points over function fields, J. Algebraic Geom. 17, (2008), 295–333. | MR 2369088 | Zbl 1221.11146

[7] P. Corvaja and U. Zannier, Greatest common divisors of u-1,v-1 in positive characteristic and rational points on curves over finite fields, J. Eur. Math. Soc. (JEMS), 15, 5 (2013), 1927–1942. | MR 3082249

[8] M. Fried and M. Jarden, Field Arithmetic, Third Edition, Springer-Verlag , New York-Heidelberg, (2008). | MR 2445111 | Zbl 1145.12001

[9] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (Fifth Ed.), Oxford Univ. Press, Oxford, (1979). | MR 568909 | Zbl 0086.25803

[10] J. Lagarias and A.M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic Number Fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, 1975), Academic Press, London, (1977) 409–464. | MR 447191 | Zbl 0362.12011

[11] F. Luca, On the greatest common divisor of u-1 and v-1 with u and v near S-units, Monatsh. Math. 146 3, (2005), 239–256. | MR 2184226 | Zbl 1107.11029

[12] W. Nöbauer, Über eine Gruppe der Zahlentheorie, Monatsh. Math. 58, (1954), 181–192. | MR 64812 | Zbl 0058.25902

[13] K. Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatsh. Math. 59, (1955), 91–97. | MR 68569 | Zbl 0064.04107

[14] J.-P. Serre, Quelques applications du theoreme de densite de Chebotarev, Publ. Math. IHES 54, (1982), 123–201. | Numdam | MR 644559 | Zbl 0496.12011

[15] J. Silverman, Common divisors of a n -1 and b n -1 over function fields, New York J. Math. 10, (2004), 37–43. | MR 2052363 | Zbl 1120.11045