Geodesic
Divisibility sequences and powers of algebraic integers
Documenta mathematica, John H. Coates' Sixtieth Birthday (2006), pp. 711-727.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $\alpha$ be an algebraic integer and define a sequence of rational integers $d_n(\alpha)$ by the condition $$ d_n(\alpha) = \max\{d\in\Bbb Z : \alpha^n \equiv 1 {MOD}{d} \}. $$ We show that $d_n(\alpha)$ is a strong divisibility sequence and that it satisfies $\log d_n(\alpha)=o(n)$ provided that no power of $\alpha$ is in $\Bbb Z$ and no power of $\alpha$ is a unit in a quadratic field. We completely analyze some of the exceptional cases by showing that $d_n(\alpha)$ splits into subsequences satisfying second order linear recurrences. Finally, we provide numerical evidence for the conjecture that aside from the exceptional cases, $d_n(\alpha)=d_1(\alpha)$ for infinitely many $n$, and we ask whether the set of such $n$ has positive (lower) density.
Classification : 11R04, 11A05, 11D61
Keywords: divisibility sequence, multiplicative group 
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     author = {Silverman, Joseph H.},
     title = {Divisibility sequences and powers of algebraic integers},
     journal = {Documenta mathematica},
     pages = {711--727},
     publisher = {mathdoc},
     volume = {John H. Coates' Sixtieth Birthday},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2006__S5__a3/}
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Silverman, Joseph H. Divisibility sequences and powers of algebraic integers. Documenta mathematica, John H. Coates' Sixtieth Birthday (2006), pp. 711-727. http://geodesic.mathdoc.fr/item/DOCMA_2006__S5__a3/