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.

For integers k2 and n0, let ν k (n) denote the greatest nonnegative integer e such that k e divides n. Moreover, let (u n ) n0 be a nondegenerate Lucas sequence satisfying u 0 =0, u 1 =1, and u n+2 =au n+1 +bu n , for some integers a and b. Shu and Yao showed that for any prime number p the sequence ν p (u n+1 ) n0 is p-regular, while Medina and Rowland found the rank of ν p (F n+1 ) n0 , where F n is the n-th Fibonacci number.

We prove that if k and b are relatively prime then ν k (u n+1 ) n0 is a k-regular sequence, and for k a prime number we also determine its rank. Furthermore, as an intermediate result, we give explicit formulas for ν k (u n ), generalizing a previous theorem of Sanna concerning p-adic valuations of Lucas sequences.

Pour tous entiers k2 et n0, soit ν k (n) le plus grand entier positif e tel que k e divise n. De plus, soit (u n ) n0 une suite de Lucas non dégénérée telle que u 0 =0, u 1 =1 et u n+2 =au n+1 +bu n , pour certains entiers a et b. Shu et Yao ont montré que, pour tout nombre premier p, la suite ν p (u n+1 ) n0 est p-régulière. Medina et Rowland ont déterminé le rang de ν p (F n+1 ) n0 , où F n est le n-ième nombre de Fibonacci.

Nous montrons que si k et b sont premiers entre eux, alors ν k (u n+1 ) n0 est une suite k-régulière. Si de plus k est un nombre premier, nous déterminons aussi le rang de cette suite. En outre, nous donnons des formules explicites pour ν k (u n ), généralisant un théorème précédent de Sanna concernant les valuations p-adiques des suites de Lucas.

Published online:
DOI: 10.5802/jtnb.1025
Classification: 11B37,  11B85,  11A99
Keywords: Lucas sequence, Fibonacci numbers, p-adic valuation, k-regular sequence, automatic sequence
     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},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1025/}
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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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