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SOYDAN, GÖKHAN

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Now showing 1 - 10 of 25
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    Publication
    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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    A modular approach to the generalized ramanujan-nagell equation
    (Elsevier, 2022-08-20) Le, Maohua; Mutlu, Elif Kizildere; Soydan, Gokhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-7651-7001; M-9459-2017
    Let k be a positive integer. In this paper, using the modular approach, we prove that if k & EQUIV; 0 (mod 4), 30 < k < 724 and 2k -1 is an odd prime power, then under the GRH, the equation x2 + (2k -1)y = kz has only one positive integer solution (x, y, z) = (k - 1, 1, 2). The above results solve some difficult cases of Terai's conjecture concerning this equation.(c) 2022 Royal Dutch Mathematical Society (KWG).
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    On the diophantine equation (5 pn 2 - 1) x
    (Honam Mathematical Soc, 2020-03-01) Kızıldere, Elif; Soydan, Gökhan; SOYDAN, GÖKHAN; Kızıldere, Elif; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; M-9459-2017; GRN-4828-2022
    Let p be a prime number with p > 3, p 3 (mod 4) and let n be a positive integer. In this paper, we prove that the Diophantine equation (5pn(2) - 1)(x) + (p(p - 5)n(2) + 1)(y) = (pn)(z) has only the positive integer solution (x; y; z) = (1; 1; 2) where pn +/- 1 (mod 5). As an another result, we show that the Diophantine equation (35n(2) - 1)(x) + (14n(2) + 1)(y) = (7n)(z) has only the positive integer solution (x, y, z) = (1; 1; 2) where n +/- 3 (mod 5) or 5 vertical bar n. On the proofs, we use the properties of Jacobi symbol and Baker's method.
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    On triangles with coordinates of vertices from the terms of the sequences {u kn} and {vkn}
    (Croatian Acad Sciences Arts, 2020-01-01) Ömür, Neşe; Soydan, Gökhan; Ulutaş, Yücel Türker; Doğru, Yusuf; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; M-9459-2017
    In this paper, we determine some results of the triangles with coordinates of vertices involving the terms of the sequences {U-kn} and {V-kn} where U-kn are terms of a second order recurrent sequence and V-kn are terms in the companion sequence for odd positive integer k, generalizing works of Cerin. For example, the cotangent of the Brocard angle of the triangle Delta(kn) iscot(Omega(Delta kn)) = Uk(2n+3) V-2k - Vk(2n+3)Uk/(-1)U-n(2k).
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    Rational points in geometric progression on the unit circle
    (Kossuth Lajos Tudomanyegyetem, 2021-01-01) Çelik, Gamze Savaş; Sadek, Mohammad; Soydan, Gökhan; Çelik, Gamze Savaş; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; EPC-6610-2022; GEK-9891-2022
    A sequence of rational points on an algebraic planar curve is said to form an r-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio r. In this work, we prove the existence of infinitely many rational numbers r such that for each r there exist infinitely many r-geometric progression sequences on the unit circle x(2) + y(2) = 1 of length at least 3.
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    On elliptic curves induced by rational diophantine quadruples
    (Japan Acad, 2022-01-01) Dujella, Andrej; Soydan, Gökhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0001-6867-5811; M-9459-2017
    In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four non-zero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups Z/2Z x Z/kZ for k = 2, 4, 6, 8, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
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    On the number of solutions of the diophantine equation x2+2a . p b = y4
    (Editura Acad Romane, 2015-01-01) Zhu, Huilin; Le, Maohua; Soydan, Gökhan; SOYDAN, GÖKHAN; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; M-9459-2017
    Let p be a fixed odd prime. In this paper, we study the integer solutions (x, y, a, b) of the equation x(2) + 2(a).p(b) = y(4), gcd(x, y) = 1, x > 0, y > 0, a >= 0, b >= 0, and we derive upper bounds for the number of such solutions.
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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.
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    Integers of a quadratic field with prescribed sum and product
    (Ars Polona-ruch, 2023-03-01) Bremner, Andrew; Soydan, Gökhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; M-9459-2017
    For given k, $ is an element of Z we study the Diophantine systemx + y + z = k, xyz =lfor x, y, z integers in a quadratic number field, which has a history in the literature. When $ = 1, we describe all such solutions; only for k = 5, 6, do there exist solutions in which none of x, y, z are rational. The principal theorem of the paper is that there are only finitely many quadratic number fields K where the system has solutions x, y, z in the ring of integers of K. To illustrate the theorem, we solve the above Diophantine system for (k, $) = (-5, 7). Finally, in the case $ = k, the system is solved completely in imaginary quadratic fields, and we give (conjecturally) all solutions when $ = k <= 100 for real quadratic fields.
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    A note on the diophantine equation x2=4pn-4pm + l2
    (Indian Nat Sci Acad, 2021-11-11) Abu Muriefah, Fadwa S.; Le, Maohua; Soydan, Gokhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.
    Let l be a fixed odd positive integer. In this paper, using some classical results on the generalized Ramanujan-Nagell equation, we completely derive all solutions (p, x, m, n) of the equation x(2) = 4p(n)-4p(m)+l(2) with l(2) < 4p(m) for any l > 1, where p is a prime, x, m, n are positive integers satisfying gcd(x, l) = 1 and m < n. Meanwhile we give a method to solve the equation with l(2) > 4p(m). As an example of using this method, we find all solutions (p, x, m, n) of the equation for l is an element of {5, 7}.