On the intersection of two distinct k -generalized Fibonacci sequences

Diego Marques

Mathematica Bohemica (2012)

  • Volume: 137, Issue: 4, page 403-413
  • ISSN: 0862-7959

Abstract

top
Let k ≥ 2 and define F ( k ) : = ( F n ( k ) ) n ≥ 0 , the k -generalized Fibonacci sequence whose terms satisfy the recurrence relation F n ( k ) = F n - 1 ( k ) + F n - 2 ( k ) + ⋯ + F n - k ( k ) , with initial conditions 0 , 0 , ⋯ , 0 , 1 ( k terms) and such that the first nonzero term is F 1 ( k ) = 1 . The sequences F : = F ( 2 ) and T : = F ( 3 ) are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation F n ( k ) = F m ( ℓ ) . In this note, we use transcendental tools to provide a general method for finding the intersections F ( k ) ∩ F ( m ) which gives evidence supporting the Noe-Post conjecture. In particular, we prove that F ∩ T = { 0 , 1 , 2 , 13 } .

How to cite

top

Marques, Diego. "On the intersection of two distinct $k$-generalized Fibonacci sequences." Mathematica Bohemica 137.4 (2012): 403-413. <http://eudml.org/doc/247189>.

@article{Marques2012,
abstract = {Let $k\ge 2$ and define $F^\{(k)\}:=(F_n^\{(k)\})_\{n\ge 0\}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^\{(k)\}=F_\{n-1\}^\{(k)\}+F_\{n-2\}^\{(k)\}+\cdots + F_\{n-k\}^\{(k)\}$, with initial conditions $0,0,\dots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^\{(k)\}=1$. The sequences $F:=F^\{(2)\}$ and $T:=F^\{(3)\}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^\{(k)\}=F_m^\{(\ell )\}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^\{(k)\}\cap F^\{(m)\}$ which gives evidence supporting the Noe-Post conjecture. In particular, we prove that $F\cap T=\lbrace 0,1,2,13\rbrace $.},
author = {Marques, Diego},
journal = {Mathematica Bohemica},
keywords = {$k$-generalized Fibonacci numbers; linear forms in logarithms; reduction method; -generalized Fibonacci numbers; Tribonacci numbers; linear forms in logarithms; reduction method},
language = {eng},
number = {4},
pages = {403-413},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the intersection of two distinct $k$-generalized Fibonacci sequences},
url = {http://eudml.org/doc/247189},
volume = {137},
year = {2012},
}

TY - JOUR
AU - Marques, Diego
TI - On the intersection of two distinct $k$-generalized Fibonacci sequences
JO - Mathematica Bohemica
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 137
IS - 4
SP - 403
EP - 413
AB - Let $k\ge 2$ and define $F^{(k)}:=(F_n^{(k)})_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{(k)}=F_{n-1}^{(k)}+F_{n-2}^{(k)}+\cdots + F_{n-k}^{(k)}$, with initial conditions $0,0,\dots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{(k)}=1$. The sequences $F:=F^{(2)}$ and $T:=F^{(3)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{(k)}=F_m^{(\ell )}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{(k)}\cap F^{(m)}$ which gives evidence supporting the Noe-Post conjecture. In particular, we prove that $F\cap T=\lbrace 0,1,2,13\rbrace $.
LA - eng
KW - $k$-generalized Fibonacci numbers; linear forms in logarithms; reduction method; -generalized Fibonacci numbers; Tribonacci numbers; linear forms in logarithms; reduction method
UR - http://eudml.org/doc/247189
ER -

References

top
  1. Agronomof, M., Sur une suite récurrente, Mathesis 4 (1914), 125-126 French. (1914) 
  2. Alekseyev, M. A., 10.1515/integ.2011.021, Integers 11 (2011), 239-259. (2011) Zbl1228.11018MR2988061DOI10.1515/integ.2011.021
  3. Back, G., Caragiu, M., The greatest prime factor and recurrent sequences, Fibonacci Q. 48 (2010), 358-362. (2010) Zbl1221.11032MR2766785
  4. Bugeaud, Y., Mignotte, M., Siksek, S., 10.4007/annals.2006.163.969, Ann. Math. 163 (2006), 969-1018. (2006) Zbl1113.11021MR2215137DOI10.4007/annals.2006.163.969
  5. Carroll, D., Jacobson, E. T., Somer, L., Distribution of two-term recurrence sequences mod p e , Fibonacci Q. 32 (1994), 260-265. (1994) MR1285757
  6. Dujella, A., Pethö, A., A generalization of a theorem of Baker and Davenport, Q. J. Math., Oxf. II Ser. 49 (1998), 291-306. (1998) Zbl0911.11018MR1645552
  7. Dresden, G. P., A simplified Binet formula for k -generalized Fibonacci numbers, Preprint, arXiv:0905.0304v1. 
  8. Erdös, P., Selfridge, J. L., 10.1215/ijm/1256050816, Illinois J. Math. 19 (1975), 292-301. (1975) Zbl0295.10017MR0376517DOI10.1215/ijm/1256050816
  9. Feinberg, M., Fibonacci-Tribonacci, Fibonacci Q. 1 (1963), 71-74. (1963) 
  10. Ferguson, D. E., An expression for generalized Fibonacci numbers, Fibonacci Q. 4 (1966), 270-273. (1966) Zbl0139.26703MR0207622
  11. Flores, I., Direct calculation of k -generalized Fibonacci numbers, Fibonacci Q. 5 (1967), 259-266. (1967) Zbl0163.03703MR0222016
  12. Gabai, H., Generalized Fibonacci k -sequences, Fibonacci Q. 8 (1970), 31-38. (1970) Zbl0211.07301MR0263734
  13. Jacobson, E. T., Distribution of the Fibonacci numbers mod 2 k , Fibonacci Q. 30 (1992), 211-215. (1992) MR1175305
  14. Kalman, D., Generalized Fibonacci numbers by matrix methods, Fibonacci Q. 20 (1982), 73-76. (1982) Zbl0472.10016MR0660765
  15. Kessler, D., Schiff, J., A combinatoric proof and generalization of Ferguson’s formula for k -generalized Fibonacci numbers, Fibonacci Q. 42 (2004), 266-273. (2004) Zbl1076.11008MR2093882
  16. Klaška, J., Tribonacci modulo p t , Math. Bohem. 133 (2008), 267-288. (2008) MR2494781
  17. Klaška, J., Tribonacci modulo 2 t and 11 t , Math. Bohem. 133 (2008), 377-387. (2008) MR2472486
  18. Klaška, J., A search for Tribonacci-Wieferich primes, Acta Math. Univ. Ostrav. 16 (2008), 15-20. (2008) Zbl1203.11020MR2498633
  19. Klaška, J., On Tribonacci-Wieferich primes, Fibonacci Q. 46/47 (2008/09), 290-297. (2008) MR2589607
  20. Klaška, J., 10.1007/s10114-010-8433-8, Acta Math. Sin. (Engl. Ser.) 26 (2010), 465-476. (2010) Zbl1238.11016MR2591606DOI10.1007/s10114-010-8433-8
  21. Klaška, J., Skula, L., The cubic character of the Tribonacci roots, Fibonacci Q. 48 (2010), 21-28. (2010) Zbl1219.11030MR2663415
  22. Klaška, J., Skula, L., Periods of the Tribonacci sequence modulo a prime p ≡ 1 ( mod 3 ) , Fibonacci Q. 48 (2010), 228-235. (2010) Zbl1217.11020MR2722219
  23. Klaška, J., Skula, L., A note on the cubic characters of Tribonacci roots, Fibonacci Q. 48 (2010), 324-326. (2010) Zbl1220.11020MR2766780
  24. Lin, P. Y., De Moivre-type identities for the Tribonacci numbers, Fibonacci Q. 26 (1988), 131-134. (1988) Zbl0641.10006MR0938586
  25. Luca, F., Fibonacci and Lucas numbers with only one distinct digit, Port. Math. 57 (2000), 243-254. (2000) Zbl0958.11007MR1759818
  26. Luca, F., 10.1216/rmjm/1181070412, Rocky Mountain J. Math. 29 (1999), 1387-1411. (1999) Zbl0978.11010MR1743376DOI10.1216/rmjm/1181070412
  27. Luca, F., Marques, D., 10.5802/jtnb.740, J. Théor. Nombres Bordx. 22 (2010), 703-718. (2010) Zbl1231.11040MR2769339DOI10.5802/jtnb.740
  28. Marques, D., Togbé, A., 10.1016/j.crma.2010.06.006, C. R. Acad. Sci. Paris I 348 (2010), 717-720. (2010) MR2671147DOI10.1016/j.crma.2010.06.006
  29. Marques, D., Togbé, A., 10.4064/cm124-2-1, Colloq. Math. 124 (2011), 145-155. (2011) Zbl1246.11036MR2842943DOI10.4064/cm124-2-1
  30. Matveev, E. M., 10.1070/IM2000v064n06ABEH000314, Izv. Math. 64 1217-1269 (2000), translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64 (2000), 125-180 (2000), 1217-1269. (2000) Zbl1013.11043MR1817252DOI10.1070/IM2000v064n06ABEH000314
  31. Mignotte, M., 10.1016/0304-3975(78)90043-9, Theor. Comput. Sci. 7 (1978), 117-121 French. (1978) Zbl0393.10009MR0498356DOI10.1016/0304-3975(78)90043-9
  32. Noe, T. D., Post, J. V., Primes in Fibonacci n -step and Lucas n -step sequences, J. Integer Seq. 8 (2005). (2005) Zbl1101.11008MR2165333
  33. Pethö, A., Fifteen problems in number theory, Acta Univ. Sapientiae Math. 2 (2010), 72-83. (2010) Zbl1201.11012MR2643936
  34. Schlickewei, H. P., Schmidt, W. M., Linear equations in members of recurrence sequences, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20 (1993), 219-246. (1993) Zbl0803.11010MR1233637
  35. Schlickewei, H. P., Schmidt, W. M., 10.4064/aa-72-1-1-44, Acta Arith. 72 (1995), 1-44. (1995) Zbl0851.11007MR1346803DOI10.4064/aa-72-1-1-44
  36. Spickerman, W. R., Binet's formula for the Tribonacci sequence, Fibonacci Q. 20 (1982), 118-120. (1982) Zbl0486.10011MR0673292
  37. Stein, S. K., 10.1307/mmj/1028998776, Michigan Math. J. 9 (1962), 399-402. (1962) Zbl0271.10008MR0154844DOI10.1307/mmj/1028998776
  38. Waddill, E. M., Some properties of a generalized Fibonacci sequence modulo m , Fibonacci Q. 16 (1978), 344-353. (1978) Zbl0394.10007MR0514322
  39. Wolfram, D. A., Solving generalized Fibonacci recurrences, Fibonacci Q. 36 (1998), 129-145. (1998) Zbl0911.11014MR1622060

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.