Turkish Journal of Mathematics
DOI
10.55730/1300-0098.3568
Abstract
We characterize the 2-adic valuation of (Pn − 1)n≥0 , where (Pn)n≥0 denotes the Padovan sequence. In addition, we use this formula to find all the Cullen and Proth numbers that are Padovan numbers. We also fully describe the 2-adic order of (Rn + 1)n≥0 , where (Rn)n≥0 denotes the Perrin sequence, and use it to find all Woodall and Proth numbers of the second kind which are Perrin numbers. As a consequence we find that 3, 5, 9, and 65 are the only Fermat numbers in the Padovan sequence; while 3 and 7 and 2 and 5 are the only numbers of Mersenne and Thâbit ibn Kurrah in the Perrin sequence respectively.
Keywords
$2$-adic valuation, Cullen number, Fermat number, mersenne number, Padovan number, perrin number, Proth number, th\^{a}bit ibn Kurrah number, Woodall number
First Page
1183
Last Page
1196
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
BRAVO, ERIC and IRMAK, NURETTİN
(2024)
"The 2-adic valuation of shifted Padovan and Perrin numbers and applications,"
Turkish Journal of Mathematics: Vol. 48:
No.
6, Article 13.
https://doi.org/10.55730/1300-0098.3568
Available at:
https://journals.tubitak.gov.tr/math/vol48/iss6/13
