We show that if {Un}n≥0 is a Lucas sequence, then the largest n suc that |Un| = Cm1Cm2⋯Cmk with 1 ≤ m1 ≤ m2 ≤⋯ ≤ mk, where Cm is the mth Catalan number satisfies n < 6500. In case the roots of the Lucas sequence are real, we have n ∈{1, 2, 3, 4, 6, 8, 12}. As a consequence, we show that if {Xn}n≥1 is the sequence of the X coordinates of a Pell equation X2 − dY2 = ±1 with a nonsquare integer d > 1, then Xn = Cm implies n = 1.

中文翻译：

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关于卢卡斯序列的成员，它们是加泰罗尼亚数的乘积

我们证明如果{ün}n≥0是卢卡斯序列，那么最大n这样|ün| = C米1C米2⋯C米ķ和1 ≤ 米1 ≤ 米2 ≤⋯ ≤ 米ķ， 在哪里C米是个米加泰罗尼亚数满足n < 6500. 如果卢卡斯数列的根是实数，我们有n ∈{1, 2, 3, 4, 6, 8, 12}. 因此，我们证明如果{Xn}n≥1是的序列X佩尔方程的坐标X2 - d是2 = ±1有一个非平方整数d > 1， 然后Xn = C米暗示n = 1.