1. Solution by brute force method

Searching the solution by brute force method

For first try, compute n = (x²-1)/y², (x²+1)/y² for x, y less than 1000,
and find the solutions for n less than 100 which does not have the square factors.

When we find the solution of x² - ny² = -1,
then we can construct the solution for x² - ny² = +1 by

X = x² + ny²
Y = 2xy

Program

 10   ' pell1
 20   dim M(100)
 30   for I=2 to 100
 40     if moeb(I)=0 then M(I)=1 else M(I)=0
 50   next I
 60   '
 70   for X=1 to 1000:X2=X^2
 80     for Y=1 to 1000:Y2=Y^2
 90       if (X2+1)@Y2=0 then W=(X2+1)\Y2 else 120
100       if or{W<1,W>=100} then 120
110       if M(W)=0 then print W;X;Y;X2-W*Y2:M(W)=1
120       if (X2-1)@Y2=0 then W=(X2-1)\Y2 else 150
130       if or{W<1,W>=100} then 150
140       if M(W)=0 then print W;X;Y;X2-W*Y2:M(W)=1
150     next Y
160   next X
170   '
180   for I=2 to 100
190     if M(I)=0 then print I;
200   next I

moeb(I) in line 40 denotes the moebius function, which returns 0 when I has square factors.

Results

n	x	y	x2 - ny2
2	1	1	-1
3	2	1	1
5	2	1	-1
6	5	2	1
7	8	3	1
10	3	1	-1
11	10	3	1
13	18	5	-1
14	15	4	1
15	4	1	1
17	4	1	-1
19	170	39	1
21	55	12	1
22	197	42	1
23	24	5	1
26	5	1	-1
29	70	13	-1
30	11	2	1
33	23	4	1
34	35	6	1
35	6	1	1
37	6	1	-1
38	37	6	1
39	25	4	1
41	32	5	-1
42	13	2	1
47	48	7	1
51	50	7	1
53	182	25	-1
55	89	12	1
57	151	20	1
58	99	13	-1
59	530	69	1
62	63	8	1
65	8	1	-1
66	65	8	1
70	251	30	1
74	43	5	-1
77	351	40	1
78	53	6	1
79	80	9	1
82	9	1	-1
83	82	9	1
85	378	41	-1
87	28	3	1
89	500	53	-1
95	39	4	1

In this primitive search, we cannot find the solutions for n equal

31, 43, 46, 61, 67, 69, 71, 73, 86, 91, 93, 94, 97

Next, expand the search range for x, y up to 10000.

10   ' pell2
20   input N:if moeb(N)=0 then 20
30   '
40   for Y=2 to 10000:Y2=Y^2
50     W=N*Y2-1:X=isqrt(W):if res=0 then print N;X;Y;W-N*Y2:end
60     W=N*Y2+1:X=isqrt(W):if res=0 then print N;X;Y;W-N*Y2:end
70   next Y

Results are following.

n	x	y	x2 - ny2
31	1520	273	1
43	3482	531	1
46	24335	3588	1
61	29718	3805	-1
67	48842	5967	1
69	7775	936	1
71	3480	413	1
73	1068	125	-1
86	10405	1122	1
91	1574	165	1
93	12151	1260	1
97	5604	569	-1

Extend the upper bound for y up to 10⁹ and try to find n=94.

94 : 2143295 , 221064 : 1

So, for all n less than 100, we could find the solutions x, y.