# On the order of appearance of product of consecutive Fibonacci numbers

Keywords:
Fibonacci number, least common multiple, the order of appearance

### Abstract

Let $F_{n}$ be the $n$th Fibonacci number. For each positive integer $m$, the order of appearance of $m$, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$ divides $F_k$. Recently, D. Marques has obtained a formula for $z(F_{n}F_{n+1})$, $z(F_{n}F_{n+1}F_{n+2})$, and $z(F_{n}F_{n+1}F_{n+2}F_{n+3})$. In this paper, we extend Marques' result to the case $z(F_{n}F_{n+1}\cdots F_{n+k})$, for every $4\leq k \leq 6$. For instance, we prove that, for $n\geq1$,\[z(F_{n}F_{n+1}F_{n+2}F_{n+3}F_{n+4}) =\begin{cases}a, & \text{if $n\equiv1,2,3,4,5,6,7,10 \pmod {12}$, or $n\equiv8,60 \pmod {72}$};\\2a, & \text{if $n\equiv9,11\pmod {12}$, or $n\equiv24,44 \pmod {72}$};\\3a, & \text{if $n\equiv12,32,36,56 \pmod {72}$}; \\6a, & \text{if $n\equiv0,20,48,68 \pmod {72}$}\end{cases}\] where $a=[n,n+1,n+2,n+3,n+4]$.

### References

1. A. Andreassian, Fibonacci sequences modulo m, Fibonacci Quart. 12 (1974), 51-65.

2. A. Benjamin and J. Quinn, The Fibonacci numbers-exposed more discretely, Math. Mag. 76 (2013), 182-192.

3. B. Farhi and D. Kane, New results on the least common multiple of consecutive integers, Proc. Amer. Math. Soc. 137 (2009), 1933-1939.

4. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics, second ed., Addison-Wesley, 1994.

5. J. H. Halton, On the divisibility properties of Fibonacci numbers, Fibonacci Quart. 4

(1996), 217-240.

6. D. Kalman and R. Mena, The Fibonacci numbers-exposed, Math. Mag. 76 (2003), 167-181.

7. N. Khaochim and P. Pongsriiam, The general case on the order of appearance of

product of consecutive Lucas numbers, Acta Math. Univ. Comenian. (N.S.) 87 (2018), 277-289.

8. T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001.

9. T. Lengyel, The order of Fibonacci and Lucas numbers, Fibonacci Quart. 33 (1995), 234-239.

10. D. Marques, Fixed points of the order of appearance in the Fibonacci sequence, Fibonacci Quart. 50 (2012), 346-351.

11. D. Marques, On the order of appearance of integers at most one away from Fibonacci numbers, Fibonacci Quart. 50 (2012), 36-43.

12. D. Marques, The order of appearance of powers of Fibonacci and Lucas numbers, Fibonacci Quart. 50 (2012), 239-245.

13. D. Marques, The order of appearance of product of consecutive Fibonacci numbers, Fibonacci Quart. 50 (2012), 132-139.

14. D. Marques, The order of appearance of the product of consecutive Lucas numbers, Fibonacci Quart. 51 (2013), 38-43.

15. K. Onphaeng and P. Pongsriiam, Subsequences and divisibility by powers of the Fibonacci numbers, Fibonacci Quart. 52 (2014), 163-171.

16. K. Onphaeng and P. Pongsriiam, The converse of exact divisibility by powers of the Fibonacci and Lucas numbers, Fibonacci Quart. 56 (2018), 296-302.

17. P. Phunphayap and P. Pongsriiam, Explicit formulas for the p-adic valuations of Fibonomial coeficients, J. Integer Seq. 21 (2018), Article 18.3.1.

18. P. Pongsriiam, Exact divisibility by powers of the Fibonacci and Lucas numbers, J. Integer Seq. 17 (2014), Article 14.11.2.

19. P. Pongsriiam, A complete formula for the order of appearance of the powers of Lucas numbers, Commun. Korean Math. Soc. 31 (2016), 447-450.

20. P. Pongsriiam, Factorization of Fibonacci numbers into products of Lucas numbers and related results, JP J. Algebra Number Theory Appl. 38 (2016), 363-372.

21. P. Pongsriiam, Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation, Commun. Korean Math. Soc. 32 (2017), 511-522.

22. P. Pongsriiam, Fibonacci and Lucas numbers which are one away from their products, Fibonacci Quart. 55 (2017), 29-40.

23. , The order of appearance of factorials in the Fibonacci sequence and certain Diophantine equations, Period. Math. Hungar. Online first version (2018), DOI

10.10071510998-018-0268-6.

24. D. W. Robinson, The Fibonacci matrix modulo m, Fibonacci Quart. 1 (1963), 29-35.

25. T. E. Stanley, A note on the sequence of Fibonacci numbers, Math. Mag. 44 (1971), 19-22.

26. , Some remarks on the periodicity of the sequence of Fibonacci numbers, Fibonacci Quart. 14 (1976), 52-54.

27. S. Vajda, Fibonacci and Lucas numbers and the golden section: Theory and applications, Dover Publications, 2007.

28. J. Vinson, The relation of the period modulo m to the rank of apparition of m in the

Fibonacci sequence, Fibonacci Quart. 1 (1963), 37-45.

29. D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532.

2. A. Benjamin and J. Quinn, The Fibonacci numbers-exposed more discretely, Math. Mag. 76 (2013), 182-192.

3. B. Farhi and D. Kane, New results on the least common multiple of consecutive integers, Proc. Amer. Math. Soc. 137 (2009), 1933-1939.

4. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics, second ed., Addison-Wesley, 1994.

5. J. H. Halton, On the divisibility properties of Fibonacci numbers, Fibonacci Quart. 4

(1996), 217-240.

6. D. Kalman and R. Mena, The Fibonacci numbers-exposed, Math. Mag. 76 (2003), 167-181.

7. N. Khaochim and P. Pongsriiam, The general case on the order of appearance of

product of consecutive Lucas numbers, Acta Math. Univ. Comenian. (N.S.) 87 (2018), 277-289.

8. T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001.

9. T. Lengyel, The order of Fibonacci and Lucas numbers, Fibonacci Quart. 33 (1995), 234-239.

10. D. Marques, Fixed points of the order of appearance in the Fibonacci sequence, Fibonacci Quart. 50 (2012), 346-351.

11. D. Marques, On the order of appearance of integers at most one away from Fibonacci numbers, Fibonacci Quart. 50 (2012), 36-43.

12. D. Marques, The order of appearance of powers of Fibonacci and Lucas numbers, Fibonacci Quart. 50 (2012), 239-245.

13. D. Marques, The order of appearance of product of consecutive Fibonacci numbers, Fibonacci Quart. 50 (2012), 132-139.

14. D. Marques, The order of appearance of the product of consecutive Lucas numbers, Fibonacci Quart. 51 (2013), 38-43.

15. K. Onphaeng and P. Pongsriiam, Subsequences and divisibility by powers of the Fibonacci numbers, Fibonacci Quart. 52 (2014), 163-171.

16. K. Onphaeng and P. Pongsriiam, The converse of exact divisibility by powers of the Fibonacci and Lucas numbers, Fibonacci Quart. 56 (2018), 296-302.

17. P. Phunphayap and P. Pongsriiam, Explicit formulas for the p-adic valuations of Fibonomial coeficients, J. Integer Seq. 21 (2018), Article 18.3.1.

18. P. Pongsriiam, Exact divisibility by powers of the Fibonacci and Lucas numbers, J. Integer Seq. 17 (2014), Article 14.11.2.

19. P. Pongsriiam, A complete formula for the order of appearance of the powers of Lucas numbers, Commun. Korean Math. Soc. 31 (2016), 447-450.

20. P. Pongsriiam, Factorization of Fibonacci numbers into products of Lucas numbers and related results, JP J. Algebra Number Theory Appl. 38 (2016), 363-372.

21. P. Pongsriiam, Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation, Commun. Korean Math. Soc. 32 (2017), 511-522.

22. P. Pongsriiam, Fibonacci and Lucas numbers which are one away from their products, Fibonacci Quart. 55 (2017), 29-40.

23. , The order of appearance of factorials in the Fibonacci sequence and certain Diophantine equations, Period. Math. Hungar. Online first version (2018), DOI

10.10071510998-018-0268-6.

24. D. W. Robinson, The Fibonacci matrix modulo m, Fibonacci Quart. 1 (1963), 29-35.

25. T. E. Stanley, A note on the sequence of Fibonacci numbers, Math. Mag. 44 (1971), 19-22.

26. , Some remarks on the periodicity of the sequence of Fibonacci numbers, Fibonacci Quart. 14 (1976), 52-54.

27. S. Vajda, Fibonacci and Lucas numbers and the golden section: Theory and applications, Dover Publications, 2007.

28. J. Vinson, The relation of the period modulo m to the rank of apparition of m in the

Fibonacci sequence, Fibonacci Quart. 1 (1963), 37-45.

29. D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532.

Published

2018-12-31

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