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On members of Lucas sequences which are products of Catalan numbers. (English) Zbl 1479.11038

Summary: We show that if \(\{ U_n \}_{n \geq 0}\) is a Lucas sequence, then the largest \(n\) suc that \(| U_n|= C_{m_1} C_{m_2}\cdots C_{m_k}\) with \(1\leq m_1\leq m_2\leq\cdots\leq m_k\), where \(C_m\) is the \(m\)-th Catalan number satisfies \(n<6500\). In case the roots of the Lucas sequence are real, we have \(n\in\{1,2,3,4,6,8,12\}\). As a consequence, we show that if \(\{ X_n \}_{n \geq 1}\) is the sequence of the \(X\) coordinates of a Pell equation \(X^2-d Y^2=\pm1\) with a nonsquare integer \(d>1\), then \(X_n= C_m\) implies \(n=1\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B65 Binomial coefficients; factorials; \(q\)-identities
11D72 Diophantine equations in many variables
11D45 Counting solutions of Diophantine equations
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[1] Bilu, Y., Hanrot, G. and Voutier, P. M., Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte, J. Reine Angew. Math.539 (2001) 75-122. · Zbl 0995.11010
[2] Laishram, S., Luca, F. and Sias, M., On members of Lucas sequences which are products of factorials, Monat. Math.193 (2020) 329-359. · Zbl 1478.11018
[3] Ljunggren, W., Über die Gleichung \(x^4-D y^2=1\), Arch. Math. Naturvid.45 (1942) 61-70. · JFM 68.0069.01
[4] Luca, F. and Odyniec, W. P., The characterisation of van Kampen-Flores complexes by means of system of Diophantine equations, Vestn. Syktyvkar. Univ. Ser. 1 Mat. Mekh. Inform.5 (2003) 5-10. · Zbl 1207.55011
[5] F. Luca, Advanced Problem H-599, Fibonacci Quart.41 (2003) 380; Solution: Fibonacci meets Catalan, Fibonacci Quart.42 (2004) 284.
[6] Luca, F. and Pappalardi, F., Members of binary recurrent sequences on lines of the Pascal triangle, Publ. Math. (Debrecen)67 (2005) 103-113. · Zbl 1090.11013
[7] Montgomery, H. L. and Vaughan, R. C., The large sieve, Mathematika20 (1973) 119-134. · Zbl 0296.10023
[8] Ramaré, O. and Rumely, R., Primes in arithmetic progression, Math. Comp.65 (1996) 397-425. · Zbl 0856.11042
[9] Sun, Q. and Yuan, P. Z., A note on the Diophantine equation \(x^4-D y^2=1\), Sichuan Daxue Xuebao34 (1997) 265-268. · Zbl 0908.11014
[10] Voutier, P., Primitive divisors of Lucas and Lehmer sequences, III, Math. Proc. Cambridge Philos. Soc.123 (1998) 407-419. · Zbl 1032.11007
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