Browsing by Author "Le, Maohua"

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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 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.
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.
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.