# On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

- (a)
- For $n\le 3$, it holds that
- (i)
- If $m\le 2r$, then either $(n,m,r,\ell )\in \left\{\right(1,4,2,1),\phantom{\rule{4pt}{0ex}}(1,60,56,5\left)\right\}$ or$$(n,m,r,\ell )=\left({\displaystyle 3,2+{\displaystyle \frac{1}{2}}\sum _{j=3}^{\ell +3}j!,1+{\displaystyle \frac{1}{2}}\sum _{j=3}^{\ell +3}j!,\ell}\right),$$
- (ii)
- If $m>2r$, then$$m<7.1(\ell +4)log(\ell +3).$$

- (b)
- For $n\ge 4$, it holds that$$n<2log((\ell +1)log(\ell +1))+105,\phantom{\rule{4pt}{0ex}}m<6(n+\ell +1)log(n+\ell )\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}r\le {\displaystyle \frac{m-2}{2}}.$$

**Theorem**

**2.**

## 2. Auxiliary Results

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Remark**

**1.**

**Lemma**

**4.**

## 3. The Proofs

#### 3.1. The Proof of Theorem 1

#### 3.1.1. The Case $n\le 3$

#### 3.1.2. The Case $n\ge 4$

#### 3.2. The Proof of Theorem 2

t[n_, r_] := t[n, r] = Which[n == 0, 0, 0 < n < r, 1, n >= r, Sum[t[n - i, r], {i, 1, r}]];

Catch[Do[{ n, m, r,l}; If[t[m,r] == Sum[Factorial[n+i], {i,0,l}], Print[{n,m,r,l}]], {l,1,5}, {n, 4, 109},{r, 2, 1638,2}, {m, r+1, 3276}]]returns $\{4,12,2,1\}$ as the only solution.

Catch[Do[{ n, m, r,l}; If[t[m,r] == Sum[Factorial[n+i], {i,0,l}], Print[{n,m,r,l}]], {l,1,5}, {n, 1, 3}, {r, 2, 1638,2}, {m, 2r+1, 3276}]]returns $\{2,6,2,1\}$ and $\{2,10,4,3\}$ as solutions. This finishes the proof.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Erdös, P.; Selfridge, J.L. The product of consecutive integers is never a power. Ill. J. Math.
**1975**, 19, 292–301. [Google Scholar] [CrossRef] - Brocard, H. Question 166. Nouv. Corresp. Math.
**1876**, 2, 287. [Google Scholar] - Ramanujan, S. Question 469. J. Indian Math. Soc.
**1913**, 5, 59. [Google Scholar] - Ramanujan, S. Collected Papers; Chelsea: New York, NY, USA, 1962. [Google Scholar]
- Berndt, B.C.; Galway, W. The Brocard–Ramanujan diophantine equation n! + 1 = m
^{2}. Ramanujan J.**2000**, 4, 41–42. [Google Scholar] [CrossRef] - Flaut, C.; Savin, D.; Zaharia, G. Some Applications of Fibonacci and Lucas Numbers. In Algorithms as a Basis of Modern Applied Mathematics. Studies in Fuzziness and Soft Computing; Hošková-Mayerová, Š., Flaut, C., Maturo, F., Eds.; Springer: Cham, Switzerland, 2021; Volume 404. [Google Scholar]
- Flaut, C.; Shpakivskyi, V.; Vlad, E. Some remarks regarding h(x)-Fibonacci polynomials in an arbitrary algebra. Chaos Solitons Fractals
**2017**, 99, 32–35. [Google Scholar] [CrossRef][Green Version] - Bugeaud, Y.; Mignotte, M.; Siksek, S. Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas powers. Ann. Math.
**2006**, 163, 969–1018. [Google Scholar] [CrossRef][Green Version] - Marques, D.; Togbé, A. Perfect powers among Fibonomial coefficients. C. R. Acad. Sci. Paris Ser. I
**2010**, 348, 717–720. [Google Scholar] [CrossRef] - Luca, F. Products of factorials in binary recurrence sequences. Rocky Mt. J. Math.
**1999**, 29, 1387–1411. [Google Scholar] [CrossRef] - Luca, F.; Stănică, P. F
_{1}F_{2}F_{3}F_{4}F_{5}F_{6}F_{8}F_{10}F_{12}= 11! Port. Math.**2006**, 63, 251–260. [Google Scholar] - Grossman, G.; Luca, F. Sums of factorials in binary recurrence sequences. J. Number Theory
**2002**, 93, 87–107. [Google Scholar] [CrossRef][Green Version] - Bollman, M.; Hernández, H.S.; Luca, F. Fibonacci numbers which are sums of three factorials. Publ. Math. Debr.
**2010**, 77, 211–224. [Google Scholar] - Luca, F.; Siksek, S. Factorials expressible as sums of at most three Fibonacci numbers. Proc. Edinb. Math. Soc.
**2010**, 53, 679–729. [Google Scholar] [CrossRef][Green Version] - Gabai, H. Generalized Fibonacci k-sequences. Fib. Quart.
**1970**, 8, 31–38. [Google Scholar] - Marques, D. On the intersection of two distinct k-generalized Fibonacci sequences. Math. Bohem.
**2012**, 137, 403–413. [Google Scholar] [CrossRef] - Bravo, J.J.; Luca, F. Coincidences in generalized Fibonacci sequences. J. Number Theory
**2013**, 133, 2121–2137. [Google Scholar] [CrossRef] - Dresden, G.P.; Du, Z. A Simplified Binet Formula for k-Generalized Fibonacci Numbers. J. Integer Seq.
**2014**, 17, 1–9. [Google Scholar] - Trojovský, P. On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes. Mathematics
**2019**, 7, 700. [Google Scholar] [CrossRef][Green Version] - Marques, D.; Lengyel, T. The 2-adic order of the Tribonacci numbers and the equation T
_{n}= m! J. Integer Seq.**2014**, 17, 14101. [Google Scholar] - Sobolewski, B. The 2-adic valuation of generalized Fibonacci sequences with an application to certain Diophantine equations. J. Number Theory
**2017**, 180, 730–742. [Google Scholar] [CrossRef][Green Version] - Young, P.T. 2-adic valuations of generalized Fibonacci numbers of odd order. Integers
**2018**, 18, A1. [Google Scholar] - Wolfram, A. Solving generalized Fibonacci recurrences. Fibonacci Quart.
**1998**, 36, 129–145. [Google Scholar] - Halton, J.H. On the divisibility properties of Fibonacci numbers. Fibonacci Quart.
**1966**, 4, 217–240. [Google Scholar] - Lengyel, T. The order of the Fibonacci and Lucas numbers. Fibonacci Quart.
**2002**, 33, 234–239. [Google Scholar] - Robinson, D.W. The Fibonacci matrix modulo m. Fibonacci Quart.
**1963**, 1, 29–36. [Google Scholar] - Vinson, J. The relation of the period modulo m to the rank of apparition of m in the Fibonacci sequence. Fibonacci Quart.
**1963**, 1, 37–45. [Google Scholar] - Marques, D. The order of appearance of product of consecutive Fibonacci numbers. Fibonacci Quart.
**2012**, 50, 132–139. [Google Scholar]

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Trojovská , E.; Trojovský, P. On Fibonacci Numbers of Order *r* Which Are Expressible as Sum of Consecutive Factorial Numbers. *Mathematics* **2021**, *9*, 962.
https://doi.org/10.3390/math9090962

**AMA Style**

Trojovská E, Trojovský P. On Fibonacci Numbers of Order *r* Which Are Expressible as Sum of Consecutive Factorial Numbers. *Mathematics*. 2021; 9(9):962.
https://doi.org/10.3390/math9090962

**Chicago/Turabian Style**

Trojovská , Eva, and Pavel Trojovský. 2021. "On Fibonacci Numbers of Order *r* Which Are Expressible as Sum of Consecutive Factorial Numbers" *Mathematics* 9, no. 9: 962.
https://doi.org/10.3390/math9090962