Browsing by Author "Szalay, Laszlo"

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    On the diophantine equation Σj=1k jFjp = Fnq
    (Masaryk Univ, Fac Science, 2018-01-01) Nemeth, Laszlo; Szalay, Laszlo; Soydan, Gökhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Anabilim Dalı.; M-9459-2017
    Let F-n denote the nth term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation F-1(p) + 2F(2)(p) + . . . + kF(k)(p) = F-n(q) in the positive integers k and n, where p and q are given positive integers. A complete solution is given if the exponents are included in the set {1, 2}. Based on the specific cases we could solve, and a computer search with p, q, k <= 100 we conjecture that beside the trivial solutions only F-8 = F-1 + 2F(2 )+ 3F(3 )+ 4F(4), F-4(2 )= F-1 + 2F(2) + 3F(3), and F-4(3) = F-1(3)+ 2F(2)(3 )+ 3F(3)(3) satisfy the title equation.
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    Resolution of the equation (3 x 1-1)(3x2-1) = (5y1-1)(5y2-1)
    (Rocky Mt Math Consortium, 2020-08-01) Liptai, Kalman; Nemeth, Laszlo; Soydan, Gökhan; Szalay, Laszlo; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; M-9459-2017
    Consider the diophantine equation (3(x1) - 1)(3(x2) - 1) = (5(y1) - 1)(5(y2) - 1) in positive integers x(1) <= x(2) and y(1) <= y(2). Each side of the equation is a product of two terms of a given binary recurrence. We prove that the only solution to the title equation is (x(1), x(2), y(1), y(2)) = (1, 2, 1, 1). The main novelty of our result is that we allow products of two terms on both sides.