Browsing by Author "SOYDAN, GÖKHAN"

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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).
A note on terai's conjecture concerning primitive pythagorean triples
(Hacettepe Üniversitesi, 2021-01-01) Le, Maohua; Soydan, Gökhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; M-9459-2017
Let f,g be positive integers such that f > g, gcd(f,g) =1 and f not equivalent to g (mod 2). In 1993, N. Terai conjectured that the equation x(2) + (f(2) - g(2))(y) = (f(2) + g(2))(z) has only one positive integer solution (x, y, z) = (2 fg, 2, 2). This is a problem that has not been solved yet. In this paper, using elementary number theory methods with some known results on higher Diophantine equations, we prove that if f = 2(r)s and g = 1, where r, s are positive integers satisfying 2 inverted iota s, r >= 2 and s < 2(r-)(1), then Terai's conjecture is true.
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}.
A note on the exponential diophantine equation A2n)x + (B2n)y = ((A2 + B2)n)z
(Croatian Mathematical Society, 2020-12-01) Le, Maohua; Soydan, Gökhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-6321-4132; M-9459-2017
Let A, B be positive integers such that. inin{A, B} > 1, gcd(A, B) = 1 and 2 vertical bar B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A > B-3/8, then the equation (A(2)n)(x) + (B(2)n)(y) = ((A(2) + B-2)n)(z) has no positive integer solutions (x, y, z) with x > z > y; if B > A(3)/6, then it has no solutions (x, y, z) with y > z > x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B 2 (mod 4) and A > B-3/8, then this equation has only the positive integer solution (x, y, z)= (1,1,1).
A note on the ternary purely exponential diophantine equation A x + Y = C z with A plus B = C 2
(Akademiai Kiado Zrt, 2020-06-01) Kızıldere, Elif; le, Maohua; Soydan, Gökhan; Kızıldere, Elif; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; M-9459-2017; GRN-4828-2022
Let l,m,r be fixed positive integers such that 2 vertical bar l, 3 lm, l > r and 3 vertical bar r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm(2) - 1,(l-r)lm(2) + 1} >30, then the equation (rlm(2) - 1)(x) + ((l - r)lm(2) + 1)(y) = (lm)(z) only the positive integer solution (x,y,z) = (1,1,2).
An elementary approach to the generalized ramanujan-nagell equation
(Indian Nat Sci Acad, 2023-01-13) Le, Maohua; Mutlu, Elif Kızıldere; Soydan, Gökhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-7651-7001; M-9459-2017
Let k be a fixed positive integer with k > 1. In this paper, using various elementary methods in number theory, we give criteria under which the equation x(2) + (2k - 1)(y) = k(z) has no positive integer solutions (x, y, z) with y is an element of {3, 5}.
Corrigendum on "the number of points on elliptic curves E : y2 = x3
(Korean Mathematical Soc, 2007-01-01) İnam, İlker; Soydan, Gökhan; SOYDAN, GÖKHAN; CANGÜL, İSMAİL NACİ; Bizim, Osman; BİZİM, OSMAN; Demirci, Musa; DEMİRCİ, MUSA; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0001-5765-1718; 0000-0002-0700-5774; M-9459-2017; ABA-6206-2020; A-6557-2018; AAH-1468-2021
In this work, authors considered a result concerning elliptic curves y(2) = x(3) + ex over F-p mod 8, given at [1]. They noticed that there should be a slight change at this result. They give counterexamples and the correct version of the result.
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.
On a class of generalized fermat equations of signature (2, 2n, 3)
(Academic Press Inc Elsevier Science, 2022-01-25) Chalupka, Karolina; Dabrowski, Andrzej; Soydan, Gokhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; M-9459-2017
We consider the Diophantine equation 7x(2) + y(2n) = 4z(3). We determine all solutions to this equation for n = 2, 3, 4 and 5. We formulate a Kraus type criterion for showing that the Diophantine equation 7x(2) + y(2p) = 4z(3) has no nontrivial proper integer solutions for specific primes p > 7. We computationally verify the criterion for all primes 7 < p < 10(9), p &NOTEQUexpressionL;13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7x(2) + y2p = 4z3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x(2) + 7y(2n) = 4z(3), determining all families of solutions for n = 2 and 3, as well as giving a (mostly) conjectural description of the solutions for n = 4 and primes n >= 5.
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.
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; Bursa Uludağ Üniversitesi/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).
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.
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.
On the exponential diophantine equation x 2+2a pb = yn
(Springer, 2015-06-01) Zhu, Huilin; Le, Maohua; Soydan, Gökhan; Togbe, Alain; SOYDAN, GÖKHAN; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0002-6321-4132; M-9459-2017
Let be an odd prime. In this paper we study the integer solutions(x, y, n, a, b) of the equation x(2) + 2(a) p(b) = y(n), x >= 1, y > 1, gcd(x, y) = 1, a >= 0, b >= 0, n >= 3.
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.
On the power values of the sum of three squares in arithmetic progression
(Univ Osijek, 2022-01-01) LE, Maohua; Soydan, Gökhan; SOYDAN, GÖKHAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; M-9459-2017
In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation (x-d)(2) +x(2)+ (x+d)(2) = y(n) (*), when n is an odd prime and d = p(r), p > 3, a prime. So this improves the results of the papers of A. Koutsianas and V. Patel [19] and A. Koutsianas [18]. Secondly, under the assumption of our first result, we prove that (*) has at most one solution (x, y). Next, for a general d, we prove the following two results: (i) if every odd prime divisor q of d satisfies q +/- 1 (mod 2n), then (*) has only the solution (x, y, d, n) = (21, 11, 2, 3), and (ii) if n > 228000 and d > 8 root 2, then all solutions (x,y) of (*) satisfy y(n) < 2(3/2)d(3).
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).
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.
Rational sequences on different models of elliptic curves
(Croatian Mathematical Soc, 2019-06-01) Sadek, Mohammad; Soydan, Gökhan; SOYDAN, GÖKHAN; Çelik, Gamze Savaş; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-6321-4132; M-9459-2017
Given a set S of elements in a number field k, we discuss the existence of planar algebraic curves over k which possess rational points whose x-coordinates are exactly the elements of S. If the size vertical bar S vertical bar of S is either 4, 5, or 6, we exhibit infinite families of (twisted) Edwards curves and (general) Huff curves for which the elements of S are realized as the x-coordinates of rational points on these curves. This generalizes earlier work on progressions of certain types on some algebraic curves.
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.