Solve Pell's equation Bonjour, How can I solve a pell's equation $x^2-181y^2=180$? I try continued fraction method but it is too long.
Get all fundamental/root/seed solutions in pari/gp:

? bnfisintnorm(bnfinit('X^2-181),180)
%1 = [-604*X + 8126, X + 19, -12*X - 162, 12*X - 162, -X + 19, -701*X + 9431]


Get all (in some limits of power of fundamental unit) solutions of source equation and ordered set of $X$:

pell()=
{
 SX= Set();
 Q= bnfinit('X^2-181);
 fu= Q.fu[1]; print("Fundamental Unit: "fu"\n");
 N= bnfisintnorm(Q, 180);  print("Fundamental Solutions: "N"\n");
 for(k=1, #N, n= N[k];
  for(l=0, 10,
   s= lift(n*fu^l);
   X= abs(polcoeff(s, 0)); Y= abs(polcoeff(s, 1));  
   if(X^2-181*Y^2==180, print1("("X","Y"), "); SX= setunion(SX, [X]))
  )
 );
 print("\n\nX= "SX)
};


Output:

? pell()
Fundamental Unit: Mod(97/2*X - 1305/2, X^2 - 181)

Fundamental Solutions: [-604*X + 8126, X + 19, -12*X - 162, 12*X - 162, -X + 19, -701*X
+ 9431]

(8126,604), (13838787971,1028629009), (23567829561880091,1751782975309639), (40136650075
266126947486,2983333705091619948244), (68353798767706678547434360031,5080697849779314462
622824949), (116408364853931064996526229733566351,8652565617013133237632070084472379), (
19,1), (4722814,351044), (8043079757959,597837410189), (13697581990952919079,10181332511
61591059), (23327351965298533840594174,1733908416325373099025404), (39727110235392768608
954621445619,2952892848528333039462775106849), (162,12), (473877,35223), (807025325517,5
9985720033), (1374385919038766082,102157300830604668), (2340616328542025659004697,173976
641561582190210003), (3986132804146565946058753351737,296286917948594475345940174413), (
162,12), (275416497,20471547), (469041730636257,34863597272757), (798790731399997433442,
59373647472611063892), (1360362182923474387351792677,101114924734303538709528327), (2316
733527296817024737161429899837,172201446925427378970399287081937), (19,1), (27634699,205
4071), (47062638533854,3498138372916), (80148944114366141359,5957424098809962661), (1364
95815848209563981659839,10145654090720536142301931), (232456059776448640174879844491294,
17278322849151565080717220682476), (9431,701), (16061239511,1193822531), (27352724540690
366,2033112003500636), (46582428416342230698371,3462444635984483802641), (79331133318570
707395013978651,5896636701084714384956793671), (135103061982078933674392128789378206,100
42131511134735440233828971339476),

X= [19, 162, 8126, 9431, 473877, 4722814, 27634699, 275416497, 13838787971, 16061239511,
 807025325517, 8043079757959, 47062638533854, 469041730636257, 23567829561880091, 273527
24540690366, 1374385919038766082, 13697581990952919079, 80148944114366141359, 7987907313
99997433442, 40136650075266126947486, 46582428416342230698371, 2340616328542025659004697
, 23327351965298533840594174, 136495815848209563981659839, 1360362182923474387351792677,
 68353798767706678547434360031, 79331133318570707395013978651, 3986132804146565946058753
351737, 39727110235392768608954621445619, 232456059776448640174879844491294, 23167335272
96817024737161429899837, 116408364853931064996526229733566351, 1351030619820789336743921
28789378206]


Verifing in Wolfram.
Get all fundamental/root/seed solutions in pari/gp:

? bnfisintnorm(bnfinit('X^2-181),180)
%1 = [-604*X + 8126, X + 19, -12*X - 162, 12*X - 162, -X + 19, -701*X + 9431]


Get all (in some limits of power of fundamental unit) solutions of source equation and ordered set of $X$:

pell()=
{
 SX= Set();
 Q= bnfinit('X^2-181);
 fu= Q.fu[1]; print("Fundamental Unit: "fu"\n");
 N= bnfisintnorm(Q, 180);  print("Fundamental Solutions: "N"\n");
 for(k=1, #N, n= N[k];
  for(l=0, 10,
   s= lift(n*fu^l);
   X= abs(polcoeff(s, 0)); Y= abs(polcoeff(s, 1));  
   if(X^2-181*Y^2==180, print1("("X","Y"), "); SX= setunion(SX, [X]))
  )
 );
 print("\n\nX= "SX)
};


Output:

? pell()
Fundamental Unit: Mod(97/2*X - 1305/2, X^2 - 181)

Fundamental Solutions: [-604*X + 8126, X + 19, -12*X - 162, 12*X - 162, -X + 19, -701*X
+ 9431]

(8126,604), (13838787971,1028629009), (23567829561880091,1751782975309639), (40136650075
266126947486,2983333705091619948244), (68353798767706678547434360031,5080697849779314462
622824949), (116408364853931064996526229733566351,8652565617013133237632070084472379), (
19,1), (4722814,351044), (8043079757959,597837410189), (13697581990952919079,10181332511
61591059), (23327351965298533840594174,1733908416325373099025404), (39727110235392768608
954621445619,2952892848528333039462775106849), (162,12), (473877,35223), (807025325517,5
9985720033), (1374385919038766082,102157300830604668), (2340616328542025659004697,173976
641561582190210003), (3986132804146565946058753351737,296286917948594475345940174413), (
162,12), (275416497,20471547), (469041730636257,34863597272757), (798790731399997433442,
59373647472611063892), (1360362182923474387351792677,101114924734303538709528327), (2316
733527296817024737161429899837,172201446925427378970399287081937), (19,1), (27634699,205
4071), (47062638533854,3498138372916), (80148944114366141359,5957424098809962661), (1364
95815848209563981659839,10145654090720536142301931), (232456059776448640174879844491294,
17278322849151565080717220682476), (9431,701), (16061239511,1193822531), (27352724540690
366,2033112003500636), (46582428416342230698371,3462444635984483802641), (79331133318570
707395013978651,5896636701084714384956793671), (135103061982078933674392128789378206,100
42131511134735440233828971339476),

X= [19, 162, 8126, 9431, 473877, 4722814, 27634699, 275416497, 13838787971, 16061239511,
 807025325517, 8043079757959, 47062638533854, 469041730636257, 23567829561880091, 273527
24540690366, 1374385919038766082, 13697581990952919079, 80148944114366141359, 7987907313
99997433442, 40136650075266126947486, 46582428416342230698371, 2340616328542025659004697
, 23327351965298533840594174, 136495815848209563981659839, 1360362182923474387351792677,
 68353798767706678547434360031, 79331133318570707395013978651, 3986132804146565946058753
351737, 39727110235392768608954621445619, 232456059776448640174879844491294, 23167335272
96817024737161429899837, 116408364853931064996526229733566351, 1351030619820789336743921
28789378206]


Verifing in Wolfram.