Browsing by Author "Soydan, Gokhan"

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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 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 ternary diophantine equation x2 - y2m = zn
(Ovidius Univ Press, 2021-01-01) Berczes, Attila; Le, Maohua; Pink, Istvan; Soydan, Gokhan; Soydan, Gokhan; Bursa Uludağ Üniversitesi/-Fen Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-6321-4132; M-9459-2017
Let N be the set of all positive integers. In this paper, using some known results on various types of Diophantine equations, we solve a couple of special cases of the ternary equation x(2) - y(2m) = z(n), x, y, z, m, n is an element of N, gcd(x, y) = 1, m >= 2, n >= 3.
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 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).