Project Euler：Problem 66 Diophantine equation

Consider quadratic Diophantine equations of the form:

x2 – Dy2 = 1
For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 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:

32 – 2×22 = 1

22 – 3×12 = 1

92 – 5×42 = 1

52 – 6×22 = 1

82 – 7×32 = 1
Hence, by considering minimal solutions in x for D ≤ 7, the largest x is obtained when D=5.
Find the value of D ≤ 1000 in minimal solutions of x for which the largest value of x is obtained.

首先要了解佩尔方程及其解法。
还要了解一些连分式的性质。

import math

maxn=1
maxD=2
for D in range(2,1001):
    tmp=math.sqrt(D)
    tmp=int(tmp)
    if tmp*tmp==D:
        continue

    m=0
    d=1
    a=tmp
    #print(a)
    n1=1
    d1=0
    num=a
    den=1

    while num*num-D*den*den!=1:
        m=int(a*d-m)
        d=int((D-m*m)/d)
        a=int((tmp+m)/d)

        n2=n1
        n1=num
        d2=d1
        d1=den

        num=int(a*n1+n2)
        den=int(a*d1+d2)
        #print('num = ',num)
        #print('den = ',den)
        #print ('result = ',num*num-D*den*den)
    if num >maxn:
        maxn=num
        maxD=D
print(maxD)