On the Diophantine Equation
نویسندگان
چکیده
In this remark, we use some properties of simple continued fractions of quadratic irrational numbers to prove that the equation x3 − 1 x− 1 = y − 1 y − 1 , x, y, n ∈ N, x > 1, y > 1, n > 3, 2 ∤ n has only the solutions (x, y, n) = (5, 2, 5) and (90, 2, 13). For any positive integer N with N > 2, let s(N) denote the number of solutions (x,m) of the equation (1) N = x − 1 x− 1 , x,m ∈ N, x ≥ 2, m > 2. Ratat [17] in 1916 and Goormaghtigh [10] in 1917 found that s(31) = 2 and s(8191) = 2, respectively. We consider the equation (2) x − 1 x− 1 = y − 1 y − 1 , x > 1, y > 1,m > 2, n > 2, x 6= y, for x, y ∈ N. It has been conjectured that the equation (2) has only a finite number of solutions, even that has only two solutions (x, y,m, n) = (5, 2, 3, 5), (90, 2, 3, 13). This is rather a difficult question. Many authors have proved that if two of the variables x, y,m, n are fixed then the equation (2) has a finite number of solutions. See for examples [1, 3, 4, 5, 12, 13, 19, 20, 21, 16, 22, 23, 24]. Remark that two known solutions of (2) are both satisfying m = 3. If m = 3, the equation (2) has the form (3) x − 1 x− 1 = y − 1 y − 1 , x, y, n ∈ N, x > y > 1, n > 3. 2000 Mathematics Subject Classification. 11A55, 11D09, 11D61.
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