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; Jack Sonn

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.

Received: 2013-09-11
Accepted: 2014-05-22
Published online: 2015-05-21

MR: 3346964

DOI: 10.5802/jtnb.893
Classification: 11A05, 11R47, 11N37
Keywords: Greatest common divisor, sequence, cyclotomic polynomial

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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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