【除了brute force还可以咋办？】x^2 – D×y^2 = 1 求解# WaterWorld - 未名水世界
C*r
1 楼
Consider quadratic Diophantine equations of the form:
x^2 – D×y^2 = 1
For example, when D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.
It can be assumed that there are no solutions in positive integers when D is
square.
By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the
following:
3^2 – 2×2^2 = 1
2^2 – 3×1^2 = 1
9^2 – 5×4^2 = 1
5^2 – 6×2^2 = 1
8^2 – 7×3^2 = 1
Hence, by considering minimal solutions in x for D < 8, the largest x is
obtained when D=5.
Find the value of D < 1001 in minimal solutions of x for which the largest
value of x is obtained.
x^2 – D×y^2 = 1
For example, when D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.
It can be assumed that there are no solutions in positive integers when D is
square.
By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the
following:
3^2 – 2×2^2 = 1
2^2 – 3×1^2 = 1
9^2 – 5×4^2 = 1
5^2 – 6×2^2 = 1
8^2 – 7×3^2 = 1
Hence, by considering minimal solutions in x for D < 8, the largest x is
obtained when D=5.
Find the value of D < 1001 in minimal solutions of x for which the largest
value of x is obtained.
