Abstract
Let (F n ) n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as
. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.
Similar content being viewed by others
References
G. Fontené, “Généralisation d’une formule connue,” Nouv. Ann. Math. 4, 112 (1915).
H. W. Gould, “The bracket function and Fontené-Ward generalized binomial coefficients with application to Fibonomial coefficients,” Fibonacci Quart. 7, 23–40 (1969).
H. W. Gould, “Generalization of Hermite’s divisibility theorems and the Mann-Shanks primality criterion for s-Fibonomial arrays,” Fibonacci Quart. 12, 157–166 (1974).
J. H. Halton, “On the divisibility properties of Fibonacci numbers,” Fibonacci Quart. 4, 217–240 (1966).
T. Koshy, Fibonacci and Lucas Numbers with Applications (Wiley, New York, 2001).
A. Kreutz, J. Lelis, D. Marques, E. Silva and P. Trojovský, “The p-adic order of the k-Fibonacci and k-Lucas numbers,” p-Adic Numbers Ultrametric Anal. Appl. 9 (1), 15–21 (2017).
T. Lengyel, “The order of the Fibonacci and Lucas numbers,” Fibonacci Quart. 33, 234–239 (1995).
D. Marques, “On integer numbers with locally smallest order of appearance in the Fibonacci sequence,” Internat. J. Math. Math. Sci. 2011, Article ID 407643, 4 pages (2011).
D. Marques, “On the order of appearance of integers at most one away from Fibonacci numbers,” Fibonacci Quart. 50, 36–43 (2012).
D. Marques, “The order of appearance of product of consecutive Fibonacci numbers,” Fibonacci Quart. 50, 132–139 (2012).
D. Marques, “The order of appearance of powers of Fibonacci and Lucas numbers,” Fibonacci Quart. 50, 239–245 (2012).
D. Marques, “Fixed points of the order of appearance in the Fibonacci sequence,” Fibonacci Quart. 51, 346–351 (2013).
D. Marques, “Sharper upper bounds for the order of appearance in the Fibonacci sequence,” FibonacciQuart. 51, 233–238 (2013).
D. Marques and P. Trojovský, “On parity of Fibonomial coefficients,” Util. Math. 101, 79–89 (2016).
D. Marques and P. Trojovský, “On divisibility properties of Fibonomial coefficients by 3,” J. Integer Seq. 15, Article 12. 6. 4, 10 pages (2012).
D. Marques, J. Sellers and P. Trojovský, “On divisibility properties of certain Fibonomial coefficients by p,” Fibonacci Quart. 51, 78–83 (2013).
D. Marques and P. Trojovský, “The p-adic order of some Fibonomial coefficients,” J. Integer Seq. 18, 15. 3. 1, 10 pages (2015).
I. Niven, H. S. Zuckermann and H. L. Montgomery, An Introduction to the Theory of Numbers (Wiley, New York, 1991).
P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (Springer-Verlag, New York, 2000).
D. W. Robinson, “The Fibonacci matrix modulo m,” Fibonacci Quart. 1, 29–36 (1963).
J. Vinson, “The relation of the period modulo m to the rank of apparition of m in the Fibonacci sequence,” Fibonacci Quart. 1, 37–45 (1963).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
Cite this article
Trojovský, P. The p-adic order of some fibonomial coefficients whose entries are powers of p . P-Adic Num Ultrametr Anal Appl 9, 228–235 (2017). https://doi.org/10.1134/S2070046617030050
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046617030050