Abstract
Let (P_{n})_{nge 0} be the sequence of Padovan numbers defined by P_0=0 , P_1 = P_2=1, and P_{n+3}= P_{n+1} +P_n for all nge 0 . In this paper, we find all positive square-free integers d such that the Pell equations x^2-dy^2 = N with Nin {pm 1, pm 4} , have at least two positive integer solutions (x, y) and (x^{prime }, y^{prime }) such that both x and x^{prime } are sums of two Padovan numbers.
Highlights
Let ðPnÞn ! 0 be the sequence of Padovan numbers defined by the linear recurrence P0 1⁄4 0; P1 1⁄4 1; P2 1⁄4 1; and Pnþ3 1⁄4 Pnþ1 þ Pn for all n ! 0: The Padovan sequence appears as sequence A000931 on the On-Line Encyclopedia of Integer Sequences (OEIS) [20]
It is well known that the Pell equations x2 À dy2 1⁄4 Æ1; ð1Þ
We extend the results from the Pell equation (1) to the Pell equation (2)
Summary
Let ðPnÞn ! 0 be the sequence of Padovan numbers defined by the linear recurrence P0 1⁄4 0; P1 1⁄4 1; P2 1⁄4 1; and Pnþ3 1⁄4 Pnþ þ Pn for all n ! 0: The Padovan sequence appears as sequence A000931 on the On-Line Encyclopedia of Integer Sequences (OEIS) [20]. 0 be the sequence of Padovan numbers defined by the linear recurrence P0 1⁄4 0; P1 1⁄4 1; P2 1⁄4 1; and Pnþ3 1⁄4 Pnþ þ Pn for all n ! 0: The Padovan sequence appears as sequence A000931 on the On-Line Encyclopedia of Integer Sequences (OEIS) [20]. The first few terms of this sequence are Supported by the Austrian Science Fund (FWF) projects: F5510-N26—Part of the special research program (SFB), ‘‘Quasi-Monte Carlo Methods: Theory and Applications’’ and W1230—‘‘Doctoral Program Discrete Mathematics’’
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