The fundamental solutions of the Diophantine equations x2 – dy2 = 1 and x2 – dy2 = 4

Let d > 1 and not a square.
Let ε4 = (t4 + u4√d)/2, where (t4,u4) is the least positive solution of x2 – dy2 = 4.
Let ε1 = t1 + u1√d, where (t1,u1) is the least positive solution of x2 – dy2 = 1.
Then B. Stolt On the Diophantine equation u2 – Dv2 = 4N, Ark. fö Mat. (1951) 1-23, stated the following:

(i) If d ≡ 1 (mod 8) or d ≡ 2 (mod 4) or d ≡ 3 (mod 4), then ε1 = ε4;
(ii) if d ≡ 5 (mod 8), then ε1 = ε4 if u4 is even, whereas if u4 is odd, then ε1 = ε43;
(iii) if d ≡ 0 (mod 4), then ε1 = ε4 if u4 is even, whereas if u4 is odd, then ε1 = ε42.

Enter d (1 < d < 1010, d nonsquare):

Last modified 15th May 2017
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