For integers and , let denote the greatest nonnegative integer such that divides . Moreover, let be a nondegenerate Lucas sequence satisfying , , and , for some integers and . Shu and Yao showed that for any prime number the sequence is -regular, while Medina and Rowland found the rank of , where is the -th Fibonacci number.
We prove that if and are relatively prime then is a -regular sequence, and for a prime number we also determine its rank. Furthermore, as an intermediate result, we give explicit formulas for , generalizing a previous theorem of Sanna concerning -adic valuations of Lucas sequences.
Pour tous entiers et , soit le plus grand entier positif tel que divise . De plus, soit une suite de Lucas non dégénérée telle que , et , pour certains entiers et . Shu et Yao ont montré que, pour tout nombre premier , la suite est -régulière. Medina et Rowland ont déterminé le rang de , où est le -ième nombre de Fibonacci.
Nous montrons que si et sont premiers entre eux, alors est une suite -régulière. Si de plus est un nombre premier, nous déterminons aussi le rang de cette suite. En outre, nous donnons des formules explicites pour , généralisant un théorème précédent de Sanna concernant les valuations -adiques des suites de Lucas.
Accepted:
Published online:
DOI: 10.5802/jtnb.1025
Keywords: Lucas sequence, Fibonacci numbers, $p$-adic valuation, $k$-regular sequence, automatic sequence
@article{JTNB_2018__30_1_227_0, author = {Nadir Murru and Carlo Sanna}, title = {On the $k$-regularity of the $k$-adic valuation of {Lucas} sequences}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {227--237}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {30}, number = {1}, year = {2018}, doi = {10.5802/jtnb.1025}, zbl = {1446.11024}, mrnumber = {3809718}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1025/} }
TY - JOUR AU - Nadir Murru AU - Carlo Sanna TI - On the $k$-regularity of the $k$-adic valuation of Lucas sequences JO - Journal de théorie des nombres de Bordeaux PY - 2018 SP - 227 EP - 237 VL - 30 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1025/ DO - 10.5802/jtnb.1025 LA - en ID - JTNB_2018__30_1_227_0 ER -
%0 Journal Article %A Nadir Murru %A Carlo Sanna %T On the $k$-regularity of the $k$-adic valuation of Lucas sequences %J Journal de théorie des nombres de Bordeaux %D 2018 %P 227-237 %V 30 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1025/ %R 10.5802/jtnb.1025 %G en %F JTNB_2018__30_1_227_0
Nadir Murru; Carlo Sanna. On the $k$-regularity of the $k$-adic valuation of Lucas sequences. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 227-237. doi : 10.5802/jtnb.1025. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1025/
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