A cyclotomic generalization of the sequence  $\gcd (a^n-1,b^n-1)$

Joseph Cohen; Jack Sonn

doi:10.5802/jtnb.893

A cyclotomic generalization of the sequence

gcd (a^{n} - 1, b^{n} - 1)

Joseph Cohen ¹; Jack Sonn ¹

¹ Department of Mathematics Technion — Israel Institute of Technology Haifa, 32000 Israel

Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 53-65.

Abstract
Résumé

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 $ℤ, ℂ [T]$ , or $𝔽_{q} [T]$ . In the case of $ℤ$ , Bugeaud, Corvaja and Zannier have obtained an upper bound $exp (ϵ n)$ 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 $exp (exp (c \frac{log n}{log log n}))$ for infinitely many $n$ , where $c$ is an absolute constant. Silverman has proved an analogous lower bound $deg gcd (a^{n} - 1, b^{n} - 1) \geq c n$ 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 $m$ th cyclotomic polynomial, Silverman’s result being the case $m = 1$ . Over $ℤ$ , 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 $ℤ$ under GRH (the generalized Riemann Hypothesis).

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 $ℤ, ℂ [T]$ ou $𝔽_{q} [T]$ , ont été étudiées depuis des décennies. Dans le cas de $ℤ$ , Bugeaud, Corvaja et Zannier ont obtenu une borne supérieure $exp (ϵ n)$ pour tout $ϵ > 0$ donné et tout $n$ grand, et montrent que la borne est optimale en extrayant la borne inférieure $exp (exp (c \frac{log n}{log log n}))$ , 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 $deg gcd (a^{n} - 1, b^{n} - 1) \geq c n$ 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 $m$ ième polynôme cyclotomique, le résultat de Silverman correspondant au cas $m = 1$ . Sur $ℤ$ , 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 $ℤ$ pour tout $m$ , sous GRH.

MR

DOI: 10.5802/jtnb.893

Classification: 11A05, 11R47, 11N37
Keywords: Greatest common divisor, sequence, cyclotomic polynomial

Author's affiliations:

Joseph Cohen ¹; Jack Sonn ¹

¹ Department of Mathematics Technion — Israel Institute of Technology Haifa, 32000 Israel

@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
AU  - Joseph Cohen
AU  - Jack Sonn
TI  - A cyclotomic generalization of the sequence  $\gcd (a^n-1,b^n-1)$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 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/
DO  - 10.5802/jtnb.893
LA  - en
ID  - JTNB_2015__27_1_53_0
ER  -

%0 Journal Article
%A Joseph Cohen
%A Jack Sonn
%T A cyclotomic generalization of the sequence  $\gcd (a^n-1,b^n-1)$
%J Journal de théorie des nombres de Bordeaux
%D 2015
%P 53-65
%V 27
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.893/
%R 10.5802/jtnb.893
%G en
%F 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, Volume 27 (2015) no. 1, pp. 53-65. doi : 10.5802/jtnb.893. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.893/

References
Cited by

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

Cited by Sources: