Previous |  Up |  Next

Article

MSC: 11B39, 11D61, 11J86
Keywords:
$k$-generalized Fibonacci numbers; linear forms in logarithms; reduction method
Summary:
Let $k\geq 2$ and define $F^{(k)}:=(F_n^{(k)})_{n\geq 0}$, 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 $F\cap T=\{0,1,2,13\}$.
References:
[1] Agronomof, M.: Sur une suite récurrente. Mathesis 4 (1914), 125-126 French.
[2] Alekseyev, M. A.: On the intersections of Fibonacci, Pell, and Lucas numbers. Integers 11 (2011), 239-259. DOI 10.1515/integ.2011.021 | MR 2988061 | Zbl 1228.11018
[3] Back, G., Caragiu, M.: The greatest prime factor and recurrent sequences. Fibonacci Q. 48 (2010), 358-362. MR 2766785 | Zbl 1221.11032
[4] Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I: Fibonacci and Lucas perfect powers. Ann. Math. 163 (2006), 969-1018. DOI 10.4007/annals.2006.163.969 | MR 2215137 | Zbl 1113.11021
[5] Carroll, D., Jacobson, E. T., Somer, L.: Distribution of two-term recurrence sequences mod $p^e$. Fibonacci Q. 32 (1994), 260-265. MR 1285757
[6] Dujella, A., Pethö, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math., Oxf. II Ser. 49 (1998), 291-306. MR 1645552 | Zbl 0911.11018
[7] Dresden, G. P.: A simplified Binet formula for $k$-generalized Fibonacci numbers. Preprint, arXiv:0905.0304v1.
[8] Erdös, P., Selfridge, J. L.: The product of consecutive integers is never a power. Illinois J. Math. 19 (1975), 292-301. MR 0376517 | Zbl 0295.10017
[9] Feinberg, M.: Fibonacci-Tribonacci. Fibonacci Q. 1 (1963), 71-74.
[10] Ferguson, D. E.: An expression for generalized Fibonacci numbers. Fibonacci Q. 4 (1966), 270-273. MR 0207622 | Zbl 0139.26703
[11] Flores, I.: Direct calculation of $k$-generalized Fibonacci numbers. Fibonacci Q. 5 (1967), 259-266. MR 0222016 | Zbl 0163.03703
[12] Gabai, H.: Generalized Fibonacci $k$-sequences. Fibonacci Q. 8 (1970), 31-38. MR 0263734 | Zbl 0211.07301
[13] Jacobson, E. T.: Distribution of the Fibonacci numbers mod $2^k$. Fibonacci Q. 30 (1992), 211-215. MR 1175305
[14] Kalman, D.: Generalized Fibonacci numbers by matrix methods. Fibonacci Q. 20 (1982), 73-76. MR 0660765 | Zbl 0472.10016
[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. MR 2093882 | Zbl 1076.11008
[16] Klaška, J.: Tribonacci modulo $p^t$. Math. Bohem. 133 (2008), 267-288. MR 2494781
[17] Klaška, J.: Tribonacci modulo $2^t$ and $11^t$. Math. Bohem. 133 (2008), 377-387. MR 2472486
[18] Klaška, J.: A search for Tribonacci-Wieferich primes. Acta Math. Univ. Ostrav. 16 (2008), 15-20. MR 2498633 | Zbl 1203.11020
[19] Klaška, J.: On Tribonacci-Wieferich primes. Fibonacci Q. 46/47 (2008/09), 290-297. MR 2589607
[20] Klaška, J.: Tribonacci partition formulas modulo $m$. Acta Math. Sin. (Engl. Ser.) 26 (2010), 465-476. DOI 10.1007/s10114-010-8433-8 | MR 2591606 | Zbl 1238.11016
[21] Klaška, J., Skula, L.: The cubic character of the Tribonacci roots. Fibonacci Q. 48 (2010), 21-28. MR 2663415 | Zbl 1219.11030
[22] Klaška, J., Skula, L.: Periods of the Tribonacci sequence modulo a prime $p\equiv 1 \pmod 3$. Fibonacci Q. 48 (2010), 228-235. MR 2722219 | Zbl 1217.11020
[23] Klaška, J., Skula, L.: A note on the cubic characters of Tribonacci roots. Fibonacci Q. 48 (2010), 324-326. MR 2766780 | Zbl 1220.11020
[24] Lin, P. Y.: De Moivre-type identities for the Tribonacci numbers. Fibonacci Q. 26 (1988), 131-134. MR 0938586 | Zbl 0641.10006
[25] Luca, F.: Fibonacci and Lucas numbers with only one distinct digit. Port. Math. 57 (2000), 243-254. MR 1759818 | Zbl 0958.11007
[26] Luca, F.: Products of factorials in binary recurrence sequences. Rocky Mountain J. Math. 29 (1999), 1387-1411. DOI 10.1216/rmjm/1181070412 | MR 1743376 | Zbl 0978.11010
[27] Luca, F., Marques, D.: Perfect powers in the summatory function of the power tower. J. Théor. Nombres Bordx. 22 (2010), 703-718. DOI 10.5802/jtnb.740 | MR 2769339 | Zbl 1231.11040
[28] Marques, D., Togbé, A.: Perfect powers among $C$-nomial coefficients. C. R. Acad. Sci. Paris I 348 (2010), 717-720. DOI 10.1016/j.crma.2010.06.006 | MR 2671147
[29] Marques, D., Togbé, A.: On terms of linear recurrence sequences with only one distinct block of digits. Colloq. Math. 124 (2011), 145-155. DOI 10.4064/cm124-2-1 | MR 2842943 | Zbl 1246.11036
[30] Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Math. 64 1217-1269 (2000), translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64 (2000), 125-180 (2000), 1217-1269. DOI 10.1070/IM2000v064n06ABEH000314 | MR 1817252 | Zbl 1013.11043
[31] Mignotte, M.: Intersection des images de certaines suites récurrentes linéaires. Theor. Comput. Sci. 7 (1978), 117-121 French. DOI 10.1016/0304-3975(78)90043-9 | MR 0498356 | Zbl 0393.10009
[32] Noe, T. D., Post, J. V.: Primes in Fibonacci $n$-step and Lucas $n$-step sequences. J. Integer Seq. 8 (2005). MR 2165333 | Zbl 1101.11008
[33] Pethö, A.: Fifteen problems in number theory. Acta Univ. Sapientiae Math. 2 (2010), 72-83. MR 2643936 | Zbl 1201.11012
[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. MR 1233637 | Zbl 0803.11010
[35] Schlickewei, H. P., Schmidt, W. M.: The intersection of recurrence sequences. Acta Arith. 72 (1995), 1-44. MR 1346803 | Zbl 0851.11007
[36] Spickerman, W. R.: Binet's formula for the Tribonacci sequence. Fibonacci Q. 20 (1982), 118-120. MR 0673292 | Zbl 0486.10011
[37] Stein, S. K.: The intersection of Fibonacci sequences. Michigan Math. J. 9 (1962), 399-402. DOI 10.1307/mmj/1028998776 | MR 0154844 | Zbl 0271.10008
[38] Waddill, E. M.: Some properties of a generalized Fibonacci sequence modulo $m$. Fibonacci Q. 16 (1978), 344-353. MR 0514322 | Zbl 0394.10007
[39] Wolfram, D. A.: Solving generalized Fibonacci recurrences. Fibonacci Q. 36 (1998), 129-145. MR 1622060 | Zbl 0911.11014
Partner of
EuDML logo