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Australian AI Institute, Level 6, 171 La Trobe St Melbourne, Australia 3000 [email protected] † RIACS, NASA Ames Research Center Moffett Field, CA 94035-1000, USA [email protected] ‡ Department of Computer Science/Intelligent Systems Program University of Pittsburgh, Pittsburgh, PA 15260, USA [email protected] # Computer Science Department/ISI, University of Southern California 4676...

The purpose of this article is to obtain some new infinite families of Toeplitz matrices, 7-matrices and generalized Pascal triangles whose leading principal minors form the Fibonacci, Lucas, Pell and Jacobsthal sequences. We also present a new proof for Theorem 3.1 in [R. Bacher. Determinants of matrices related to the Pascal triangle. J. Théor. Nombres Bordeaux, 14:19–41, 2002.].

In this work we look at an approach to solving Pell’s equation using continued fractions and fundamental units in real quadratic orders. We demonstrate that there is an underlying general approach using Lucas-Lehmer methods for solving Pell and other quadratic Diophantine equations that is often overlooked in the literature. Mathematics Subject Classification: Primary: 11D09; 11A55; Secondary: ...

In [7] we have studied the equation m − n = py, where p is a prime natural number p ≥ 3. Using the above result, in this paper, we study the equations ck(x 4 + 6px y +py) + 4pdk(x y + pxy) = 32z with p ∈ {5, 13, 29, 37}, where (ck, dk) is a solution of the Pell equation ∣∣c2 − pd2∣∣ = 1.