PDF(461 KB)
PDF(461 KB)
PDF(461 KB)
丢番图方程X2-(a2+1)Y4 = 3-4a
On the Diophantine Equation X2-(a2+1)Y4 = 3-4a
设a ≥ 2是正整数. 本文证明了:当a=2时,方程X2-(a2+1)Y4 = 3-4a仅有正整数解(X,Y)=(20,3);当a=3时,该方程仅有2组互素的正整数解(X,Y)=(1,1)和(79,5);当a ≥4且4a+1非平方数时,该方程最多有4组互素的正整数解(X,Y);当a ≥4且4a+1为平方数时,该方程最多有5组互素的正整数解(X,Y).
Let a ≥ 2 be a positive integer. In this paper, we will prove that if a = 2, then the equation X2-(a2+1)Y4 = 3-4a has only one positive integer solution (X, Y) = (20, 3); if a = 3, then the equation has only two coprime positive integer solutions (X, Y) = (1, 1), (79, 5); if a ≥ 4 and 4a + 1 is a nonsquare positive integer, then the equation has at most four coprime positive integer solutions (X, Y); if a ≥ 4 and 4a + 1 is a square, then the equation has at most five coprime positive integer solutions (X, Y).
四次方程 / 虚二次域 / 丢番图逼近 / 解数 / 上界 {{custom_keyword}} /
quartic equations / imaginary quadratic fields / Diophantine approximations / number of positive integer solutions / upper bound {{custom_keyword}} /
[1] Cao Z. F., Introduction to Diophantine Equations (in Chinese), Harbin Institute of Technology Press, Harbin, 1989: 260-261.
[2] Dujella A., Continued fractions and RSA with small secret exponent. Tatra Mt. Math. Publ., 2004, 2: 101- 112.
[3] Hua L. K., Introduction to Number Theory, Translated from Chinese by Peter Shiu, Springer-Verlag, Berlin, New York: 1982.
[4] Ljunggren W., Zur Theorie der Gleichung x2 + 1 = Dy4, Avh. Norsk. Vid. Akad. Oslo I, 1942, 5: 1-27.
[5] Ljunggren W., On the Diophantine equation Ax4-By2 = C(C = 1, 4), Math. Scand., 1967, 21(2): 149-158.
[6] Stoll M., Walsh P. G., Yuan P. Z., On the Diophantine equation X2 - (22m + 1)Y4 = -22m, Acta Arith., 2009, 139(1): 57-63.
[7] Walsh G., On the number of large integer points on elliptic curves, Acta Arith., 2009, 138(4): 317-327.
[8] Yuan P. Z., Zhang Z. F., On the Diophantine equation X2 - (1 + a2)Y4 = -2a, Sci. China Ser. A, 2010, 53(8): 2143-2158.
[9] Yuan P. Z., Zhang Z. F., On the Diophantine equation X2 - (a2 + 4p2n)Y4 = -4p2n, Acta Mathematica Sinica, Chinese Series, 2014, 57(2): 209-222.
江苏省教育科学十二五规划课题(D201301083);云南省教育厅科研课题(2014Y462);泰州学院重点课题(TZXY2014ZDKT007)
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