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Lucas sequences and quadratic orders

机译:卢卡斯序列和二次阶

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We consider the Lucas sequences (U_n)_n ≥ 0 defined by U_0 = 0, U_1 = 1, and U_n = PU_(n-1) - QU_(n-2) for non-zero integral parameters P, Q such that Δ = P~2 - 4Q is not a square. We use the arithmetic of the quadratic order with discriminant Δ to investigate the zeros and the period length of the sequence (U_n)_(n≥0) modulo a positive integer d coprime to Q. For a prime p not dividing Q, we give precise formulas for p-powers, we determine the p-adic value of U_n, and we connect the results with class number relations for quadratic orders.
机译:对于非零积分参数P,Q,我们考虑由U_0 = 0,U_1 = 1和U_n = PU_(n-1)-QU_(n-2)定义的卢卡斯序列(U_n)_n≥0,使得Δ= P〜2-4Q不是平方。我们使用带有判别式Δ的二次数算法研究零和序列(U_n)_(n≥0)的周期长度,将正整数d互质数模为Q。对于不分解Q的质数p,我们给出精确的p幂公式,我们确定U_n的p-adic值,并将结果与​​二次序的类数关系联系起来。

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