Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
Fibonacci sequence; Diophantine equation
Fibonacci sequence; Diophantine equation
Summary:
Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots +kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\lbrace 1,2\rbrace $. Based on the specific cases we could solve, and a computer search with $p,q,k\le 100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation.
References:
[1] Alvarado, S.D., Dujella, A., Luca, F.: On a conjecture regarding balancing with powers of Fibonacci numbers. Integers 12 (2012), 1127–1158. DOI 10.1515/integers-2012-0032 | MR 3011553
[2] Andreescu, T., Andrica, D.: Quadratic Diophantine Equations. 2015, 124–126. MR 3362224
[3] Behera, A., Liptai, K., Panda, G.K., Szalay, L: Balancing with Fibonacci powers. Fibonacci Quart. 49 (2011), 28–33. MR 2781575
[4] Chaves, A.P., Marques, D., Togbé, A.: On the sum of powers of terms of a linear recurrence sequence. Bull. Braz. Math. Soc. New Series 43 (2012), 397–406. DOI 10.1007/s00574-012-0018-y | MR 3024062
[5] Koshy, T.: Fibonacci and Lucas Numbers with Applications. John Wiley and Sons, 2011. MR 1855020
[6] Luca, F., Oyono, R.: An exponential Diophantine equation related to powers of two consecutive Fibonacci numbers. Proc. Japan Acad. Ser. A 87 (2011), 45–50. MR 2803898
[7] Luca, F., Szalay, L.: Fibonacci diophantine triples. Glas. Mat. Ser. III 43 (63) (2008), 253–264. DOI 10.3336/gm.43.2.03 | MR 2460699
[8] Marques, D., Togbé, A.: On the sum of powers of two consecutive Fibonacci numbers. Proc. Japan Acad. Ser. A 86 (2010), 174–176. MR 2779831
[9] Panda, G.K.: Sequence balancing and cobalancing numbers. Fibonacci Quart. 45 (2007), 265–271. MR 2438198
[10] Pongsriiam, P.: Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation. Commun. Korean Math. Soc. 91 (3) (2017), 511–522. MR 3682410
[11] Pongsriiam, P.: Fibonacci and Lucas numbers which are one away from their products. Fibonacci Quart. 55 (2017), 29–40. MR 3620575
[12] Soydan, G.: On the Diophantine equation $(x+1)^k+(x+2)^k+\dots +(lx)^k=y^n$. Publ. Math. Debrecen 91 (3–4) (2017), 369–382. MR 3744801
[13] Vorob’ev, N.N.: Fibonacci Numbers. Blaisdell Pub. Co. New York, 1961.
[14] Wulczyn, G.: Problem E2158. Amer. Math. Monthly 76 (1969), 1144–1146. DOI 10.2307/2317203 | MR 1535701
Search
Partner of
