Browsing by Author "Soydan, Gökhan"

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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 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}.
An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples
(Springer, 2019-07-11) Le, Maohua; Soydan, Gökhan; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-6321-4132; 23566953200
Let m, n be positive integers such that m > n, gcd(m,n) = 1 and m not equivalent to n(mod2) . In 1956, L. Jesmanowicz conjectured that the equation (m(2)-n(2))(x) + (2mn)(y) = (m(2) + n(2))(z) has only the positive integer solution (x,y,z)=(2,2,2). This conjecture is still unsolved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent with some elementary methods, we prove that if mn equivalent to 2(mod 4) and m > 30.8n, then Jesmanowicz' conjecture is true.
Ardışık kuvvet toplamları ve Bernoulli polinomları
(Uludağ Üniversitesi, 2016-01-07) Savaş, Gamze; Soydan, Gökhan; Uludağ Üniversitesi/Fen Bilimleri Enstitüsü/Matematik Anabilim Dalı.
Bu çalısmanın amacı (x + 1)k + (x + 2)k + . . . + (2x)k = yn Diophant denkleminin pozitif tamsayı çözümleri için n'ye üst sınırlar bulmak ve bu üst sınırlara bağlı olarak denklemin çözümlerinin olduğu durumları belirlemektir. Tez üç ̧ bölümden oluşmaktadır. Birinci bölümde Bernoulli sayıları ve Bernoulli polinomları hakkında temel bilgiler verilmiştir. ikinci bölümde ardışık kuvvet toplamları tipindeki Diophant denklemler ve bu denklemlerle ilgili literatür bilgisi verilmiştir. Üçüncü bölümde ise (x + 1)k + (x + 2)k + ... + (2x)k = yn Diophant denkleminde k ve x'e bağlı yapılan sınıflandırmayla n için sabit üst sınırlar ile 2-sel ve 3-sel değerlendirme fonksiyonlarına bağlı üst sınırlar elde edilmiştir.
Cebirsel eğriler üzerindeki rasyonel diziler
(Bursa Uludağ Üniversitesi, 2022-01-17) Çelik, Gamze Savaş; Soydan, Gökhan; Bursa Uludağ Üniversitesi/Fen Bilimleri Enstitüsü/Matematik Anabilim Dalı.; 0000-0002-6609-1713
Tez yedi bölümden olusmaktadır. Ilk üç bölümde cebirsel ve eliptik egriler ile ilgili temel bilgiler ve bazı önemli teoremlere yer verilmistir. K; L 2 Q iken Q’da y2 = x3 + Kx + L ile verilen E eliptik egrisi olsun. i =1; : : : ; k iken noktaların x-bilesenleri xi’lerardı sık küplerden olusursa (xi; yi) 2 E(Q)rasyonel noktalar kümesinin E üzerinde ardısık küplerin bir dizisi oldugu söylenir.Tezindördüncü bölümünde ardısık küplerin 5-terimli dizilerini içeren eliptik egrilerin sonsuz bir ailesinin varlıgını gösteriyoruz. Ayrıca bu bes rasyonel noktanın E(Q)’da lineer bagımsız dolayısıyla E(Q)’nun rankı en az 5 oldugunu gösterdik. Tezin besinci bölümünde, bir F sayı cismindeki elemanların bir S alt kümesi verildiginde x-bilesenleri S’nin elemanları olan rasyonel noktalara sahip F cismi üzerindeki düzlem cebirsel egrilerin varlıgını tartısıyoruz. S-dizisinin eleman sayısı jSj = 4; 5 veya 6 iken üzerindeki rasyonel noktaların x-bilesenlerinin S’de bulundugu (bükülmüs) Edwardse grileri ve (genel) Huff egrilerinin sonsuz ailelerini sergiliyoruz. Bu, bazı cebirsel egriler üzerindeki belirli tipteki diziler hakkında yapılmıs önceki çalısmaları geneller. Bir düzlem cebirsel egri üzerindeki rasyonel noktaların x veya y-bilesenleri ortak çarpanı r olacak sekilde bir geometrik dizi olusturursa bu egri üzerindeki rasyonel noktaların dizisi bir r-geometrik dizisi olarak adlandırılır. Tezin altıncı bölümünde x2 + y2 = 1 birim çember denklemi üzerinde en az 3-terimli r-geometrik dizilerini bulunduran sonsuz çoklukta r-rasyonel sayısının varlıgını ispatlıyoruz. Son bölümde tezdeki sonuçlar tartısılmıstır ve tez sonrası gelecek çalısmalardan bahsedilmistir. çember denklemi üzerinde en az 3-terimli r-geometrik dizilerini bulunduran sonsuz çoklukta r-rasyonel sayısının varlıgını ispatlıyoruz. Son bölümde tezdeki sonuçlar tartısılmıstır ve tez sonrası gelecek çalısmalardan bahsedilmistir.
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.
Elliptic curves containing sequences of consecutive cubes
(Rocky Mountain Mathematics Consortium, 2018) Çelik, Gamze Savaş; Soydan, Gökhan; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 57206274023; 23566953200
Let E be an elliptic curve over Q described by y(2) = x(3)+Kx+L, where K, L is an element of Q. A set of rational points (x(i), y(i)) is an element of E(Q) for i = 1, 2,..., k, is said to be a sequence of consecutive cubes on E if the x-coordinates of the points x(i)'s for i = 1, 2,..., form consecutive cubes. In this note, we show the existence of an infinite family of elliptic curves containing a length-5-term sequence of consecutive cubes. Moreover, these five rational points in E(Q) are linearly independent, and the rank r of E(Q) is at least 5.
The group structure of bachet elliptic curves over finite fields f-p
(Univ Miskolc Inst Math, 2009) İkikardeş, Nazlı Yıldız; Demirci, Musa; Soydan, Gökhan; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017
Bachet elliptic curves are the curves y(2) = x(3) + a(3) and, in this work, the group structure E(F-p) of these curves over finite fields F-p is considered. It is shown that there are two possible structures E(F-p) congruent to Cp+1 or E(F-p) congruent to C-n x C-nm, for m, n is an element of N; according to p equivalent to 5 (mod 6) and p equivalent to 1 (mod 6), respectively. A result of Washington is restated in a more specific way saying that if E(F-p) congruent to Z(n) x Z(n) then p equivalent to 7 (mod 12) p = n(2) -/+ n + 1.
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.
Note on "On the Diophantine equation nx2 + 22m = y n" [Y. Wang, T. Wang, J. Number Theory 131 (8) (2011) 1486-1491]
(Elsevier, 2013-12-13) Soydan, Gökhan; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; ABA-6206-2020; 23566953200; 57189022403
This note presents corrections to the paper by Y. Wang and T. Wang (2011) [2]. The unique theorem given in that paper states that for any odd integer n > 1, nx(2) + 2(2m) = y(n) has no positive integer solution (x, y, m) with gcd(x,y) = 1. (C) 2014 Elsevier Inc. All rights reserved.
On a class of Lebesgue-Ljunggren-Nagell type equations
(Academic Press Elsevier Science, 2019-12-10) Dąbrowski, Andrzej; Günhan, Nursena; Soydan, Gökhan; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-1919-2431; HNT-0160-2023; HOC-4413-2023; 57214758192; 23566953200
Text. Given odd, coprime integers a, b (a > 0), we consider the Diophantine equation ax(2) + b(2l) = 4y(n), x, y is an element of Z, l is an element of N, n odd prime, gcd(x, y) = 1. We completely solve the above Diophantine equation for a is an element of {7, 11, 19, 43, 67, 163}, and b a power of an odd prime, under the conditions 2(n-1)b(l) not equivalent to +/- 1(mod a) and gcd (n, b) = 1. For other square-free integers a > 3 and b a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers x, y with (gcd(x, y) = 1), l is an element of N and all odd primes n > 3, satisfying 2(n-1)b(l) not equivalent to +/- 1(mod a), gcd(n, b) = 1, and gcd(n, h(-a)) = 1, where h(-a) denotes the class number of the imaginary quadratic field Q(root-a). Video. For a video summary of this paper, please visit https://youtu.be/Q0peJ2GmqeM.
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 Diophantine equation ((c+1)m2+1)x + (cm2-1)y = (am)z
(Bilimsel ve Teknolojik Araştırma Kurumu, 2018-08-10) Miyazaki, Takafumi; Kızıldere, Elif; Soydan, Gökhan; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-6321-4132; 0000-0002-7651-7001; M-9459-2017; 57204173004; 23566953200
Suppose that c, in, and a are positive integers with a 11, 13 (mod 24) . In this work, we prove that when 2c + 1 = a(2), the Diophantine equation in the title has only solution (x, y, z) = (1,1,2) where m +/- 1 (mod a) and m > a(2) in positive integers. The main tools of the proofs are elementary methods and Baker's theory.
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 (x+1)(k) (x+2)(k) + . . . plus (lx)(k) = y(n)
(Kossuth Lajos Tudomanyegyetem, 2017) Soydan, Gökhan; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 23566953200
Let k, l >= 2 be fixed integers. In this paper, firstly, 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 satisfy n < C-1, where C-1 = C-1(l, k) is an effectively computable constant. Secondly, we prove that all solutions of this equation in integers x, y, n with x,y >= 1,n >= 2, k not equal 3 and I 0 (mod 2) satisfy max{x, y, n} < C-2, where C-2 is an effectively computable constant depending only on k and I.
On the Diophantine equation (x+1)(k) + (x+2)(k) + ... + (2x)(k) = y(n)
(Elsevier, 2017-07-12) Bérczes, Attila; Pink, István; Savaş, Gamze; Soydan, Gökhan; Uludağ Üniversitesi/Fen-Edebiyet Fakültesi/Matematik Bölümü.; FWV-5620-2022; GEK-9891-2022; 57206274023; 23566953200
In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T-k(x) = (x + 1)(k) + (x + 2)(k) + ... + (2x)(k). Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. [6]), we show that for 2 <= x <= 13, k >= 1,y >= 2 and n >= 3 the title equation has no solutions.
On the diophantine equation x 2+2 a • 3 b • 11 c = y n
(Walter De Gruyter, 2013-06) Luca, Florian; Naci Cangül, İsmail; Demirci, Musa; İnam, İlker; Soydan, Gökhan; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; 0000-0002-6439-8439; 0000-0002-6321-4132; J-3505-2017; ABA-6206-2020; DUZ-5808-2022; GEK-9891-2022; 57189022403; 23566581100; 25925069700; 23566953200
In this note, we find all the solutions of the Diophantine equation x (2) + 2 (a) center dot 3 (b) center dot 11 (c) = y (n) , in nonnegative integers a, b, c, x, y, n a parts per thousand yen 3 with x and y coprime.
On the Diophantine Equation x(2) 5(a) . 11(b) = y(n)
(Wydawnictwo Naukowe, 2010) Tzanakis, Nikos; Soydan, Gökhan; Kaczorowski, J.; Cangül, İsmail Naci; Demirci, Musa; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; J-3505-2017; 57189022403; 23566581100
We give the complete solution (n, a, b, x, y) of the title equation when gcd(x,y) = 1, except for the case when xab is odd. Our main result is Theorem 1.