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SOYDAN, GÖKHAN

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Now showing 1 - 10 of 25
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    Publication
    On the conjecture of jesmanowicz
    (Centre Environment Social & Economic Research Publ-ceser, 2017-01-01) Togbe, Alain; Cangül, İsmail Naci; CANGÜL, İSMAİL NACİ; Soydan, Gokhan; SOYDAN, GÖKHAN; Demirci, Musa; DEMİRCİ, MUSA; Fen Edebiyat Fakültesi; Matematik Bölümü; 0000-0002-0700-5774; 0000-0002-5882-936X; J-3505-2017; A-6557-2018; ABA-6206-2020; M-9459-2017
    We give a survey on some results covering the last 60 years concerning Jesmanowicz' conjecture. Moreover, we conclude the survey with a new result by showing that the special Diophantine equation(20k)(x) + (99k)(y) = (101k)(z)has no solution other than (x, y, z) = (2, 2, 2).
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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; 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}.
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    Publication
    Integers of a quadratic field with prescribed sum and product
    (Ars Polona-ruch, 2023-03-01) Bremner, Andrew; Soydan, Gökhan; SOYDAN, GÖKHAN; 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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    Publication
    A modular approach to the generalized ramanujan-nagell equation
    (Elsevier, 2022-08-20) Le, Maohua; Mutlu, Elif Kizildere; Soydan, Gokhan; SOYDAN, GÖKHAN; 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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    Publication
    On the diophantine equation Σj=1k jFjp = Fnq
    (Masaryk Univ, Fac Science, 2018-01-01) Nemeth, Laszlo; Szalay, Laszlo; Soydan, Gökhan; SOYDAN, GÖKHAN; Fen Edebiyat Fakültesi; Matematik Ana Bilim 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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    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; 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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    Publication
    The Diophantine equation (x+1)k + (x+2)k + ... plus (lx)k = yn revisted
    (Univ Debrecen, Inst Mathematics, 2020-01-01) Bartoli, Daniele; Soydan, Gökhan; SOYDAN, GÖKHAN; Fen Edebiyat Fakültesi; Matematik Bölümü; 0000-0002-5767-1679; M-9459-2017
    Let k,l >= 2 be fixed integers, and C be an effectively computable constant depending only on k and l. In this paper, we prove that all solutions of the equation (x + 1)(k) + (x + 2)(k) + ... + (lx)(k) = y(n) in integers x, y,n with x, y >= 1, n >= 2, k not equal 3 and l 1 (mod 2) satisfy max{x, y, n} < C. The case when is even has already been completed by the second author (see [24]).
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    Publication
    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; 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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    Publication
    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; 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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    Publication
    On elliptic curves induced by rational diophantine quadruples
    (Japan Acad, 2022-01-01) Dujella, Andrej; Soydan, Gökhan; SOYDAN, GÖKHAN; Fen Edebiyat Fakültesi; Matematik Ana Bilim 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.