¿Que $r\in \mathbb Q$, entonces existen $a,b,c,d \in \mathbb Q$ tal que $r=\frac {a^4-b^4}{c^4-d^4} $?
Respuesta
¿Demasiados anuncios?
Oleg567
Puntos
9849
Si reescribir $r$
$$
r = \dfrac{p}{q},\etiqueta{1}
$$
donde $p,q\in\mathbb{N}$, $\quad p\le q$, $\quad GCD(p,q)=1$,
a continuación, para $p,q\le 130$ todas las fracciones $(1)$ tienen soluciones con $\max\{a,c\} \le 120000$.
(así que esto es sólo parcial de la solución del problema, por supuesto, que cubre el cuadrado de $130\times 130$ dominio $\{(p,q):\; p,q\in\mathbb{N}^2\}$).
Primeros candidatos para los contraejemplos son $\dfrac{24}{131}$, $\dfrac{73}{142}$, $\dfrac{113}{151}$, $\dfrac{118}{151}$, $\ldots$.
Si tienen soluciones $\color{#DDDDDD}{(\mbox{and I'm becoming more and more convinced: they have})}$,
a continuación, estas soluciones tienen $\max\{a,c\}>55000$.
Un par de "duro" ejemplos (por más simple $r$ y de un gran $\max\{a,c\}$):
$$\dfrac{1}{2} = \dfrac{116^4-34^4}{139^4-61^4};$$ $$\dfrac{1}{8} = \dfrac{124^4-22^4}{257^4-223^4};$$ $$\dfrac{4}{5} = \dfrac{484^4-478^4}{215^4-7^4};$$ $$\dfrac{31}{36} = \dfrac{7259^4-6763^4}{5310^4-390^4};$$ $$\dfrac{30}{67} = \dfrac{8267^4-2821^4}{10151^4 - 4255^4};$$ $$\dfrac{43}{71} = \dfrac{15847^4-13909^4}{14344^4 - 854^4};$$ $$\dfrac{79}{83} = \dfrac{53629^4-23159^4}{53817^4 - 1029^4};$$ $$\dfrac{43}{67} = \dfrac{111359^4-71993^4}{118701^4-29637^4}.$$
Here is list of smallest solutions (with smallest $\max\{a,c\}$) escrito en el formulario $$p/q: (a,b), (c,d);$$ donde $q\le 50$:
1/1: (134,59), (158,133); <-- example of nontrivial solution
1/2: (116,34), (139,61);
1/3: (3,1), (4,2);
2/3: (39,31), (38,4);
1/4: (6,4), (9,7);
3/4: (29,11), (31,1);
1/5: (19,17), (22,4);
2/5: (28,24), (29,3);
3/5: (5,1), (6,4);
4/5: (254,154), (259,13);
1/6: (7,3), (13,11);
5/6: (81,67), (73,31);
1/7: (8,2), (13,1);
2/7: (9,7), (11,3);
3/7: (11,7), (13,1);
4/7: (53,27), (68,54);
5/7: (19,17), (16,2);
6/7: (7,1), (8,6);
1/8: (139,61), (232,68);
3/8: (13,11), (14,6);
5/8: (7,1), (8,4);
7/8: (11,3), (12,8);
1/9: (17,9), (29,11);
2/9: (11,3), (16,2);
4/9: (14,6), (17,1);
5/9: (19,17), (18,12);
7/9: (8,1), (9,6);
8/9: (32,4), (33,9);
1/10: (4,2), (7,1);
3/10: (136,124), (137,9);
7/10: (10,4), (13,11);
9/10: (10,8), (9,1);
1/11: (14,12), (21,1);
2/11: (61,51), (79,9);
3/11: (13,1), (18,4);
4/11: (19,13), (23,1);
5/11: (23,14), (28,17);
6/11: (79,47), (89,23);
7/11: (19,9), (21,1);
8/11: (122,118), (79,31);
9/11: (41,23), (42,2);
10/11: (215,175), (191,59);
1/12: (17,9), (31,1);
5/12: (9,7), (10,2);
7/12: (59,53), (52,16);
11/12: (227,147), (226,122);
1/13: (3,1), (6,4);
2/13: (59,53), (73,31);
3/13: (2,1), (3,2);
4/13: (6,2), (9,7);
5/13: (4,2), (5,1);
6/13: (107,13), (131,57);
7/13: (17,11), (19,7);
8/13: (64,4), (77,53);
9/13: (11,7), (14,12);
10/13: (64,52), (83,77);
11/13: (18,4), (19,9);
12/13: (8,4), (9,7);
1/14: (6,4), (11,3);
3/14: (49,41), (61,19);
5/14: (29,3), (38,18);
9/14: (201,171), (187,61);
11/14: (35,13), (37,5);
13/14: (87,19), (89,33);
1/15: (9,8), (16,13);
2/15: (5,3), (8,2);
4/15: (67,13), (98,64);
7/15: (19,9), (23,11);
8/15: (6,2), (7,1);
11/15: (12,10), (11,1);
13/15: (59,19), (61,11);
14/15: (18,10), (19,13);
1/16: (2,1), (4,2);
3/16: (2,1), (3,1);
5/16: (11,2), (19,17);
7/16: (13,1), (16,4);
9/16: (29,11), (34,18);
11/16: (17,16), (13,7);
13/16: (3,2), (3,1);
15/16: (16,13), (18,16);
1/17: (2,1), (4,1);
2/17: (11,3), (19,9);
3/17: (14,8), (21,1);
4/17: (9,7), (14,12);
5/17: (38,34), (41,23);
6/17: (13,11), (18,16);
7/17: (26,16), (37,31);
8/17: (19,13), (31,29);
9/17: (9,3), (11,7);
10/17: (7,1), (8,2);
11/17: (43,23), (47,13);
12/17: (22,14), (23,7);
13/17: (37,23), (38,4);
14/17: (11,3), (14,12);
15/17: (8,1), (11,10);
16/17: (4,2), (4,1);
1/18: (16,2), (33,9);
5/18: (11,7), (15,9);
7/18: (31,25), (35,19);
11/18: (323,271), (321,201);
13/18: (32,22), (39,33);
17/18: (29,3), (38,34);
1/19: (47,13), (98,16);
2/19: (38,14), (67,29);
3/19: (16,13), (22,3);
4/19: (59,43), (101,89);
5/19: (42,32), (53,15);
6/19: (71,55), (85,29);
7/19: (41,29), (49,11);
8/19: (29,3), (36,2);
9/19: (39,21), (46,8);
10/19: (121,47), (143,67);
11/19: (16,6), (21,17);
12/19: (23,7), (39,37);
13/19: (34,27), (67,66);
14/19: (29,13), (31,7);
15/19: (73,43), (75,1);
16/19: (47,13), (49,8);
17/19: (118,38), (121,7);
18/19: (237,171), (226,116);
1/20: (127,77), (259,13);
3/20: (5,1), (9,7);
7/20: (31,17), (40,20);
9/20: (27,21), (38,34);
11/20: (93,83), (84,16);
13/20: (47,31), (56,44);
17/20: (39,37), (27,11);
19/20: (32,6), (33,17);
1/21: (18,8), (41,29);
2/21: (9,1), (17,11);
4/21: (13,3), (23,19);
5/21: (7,3), (10,4);
8/21: (64,4), (83,43);
10/21: (71,17), (87,45);
11/21: (7,4), (8,1);
13/21: (14,12), (13,1);
16/21: (36,16), (41,29);
17/21: (41,27), (52,46);
19/21: (79,3), (85,55);
20/21: (30,10), (31,17);
1/22: (61,59), (79,31);
3/22: (14,4), (23,1);
5/22: (23,11), (37,29);
7/22: (347,165), (476,300);
9/22: (64,62), (51,37);
13/22: (17,9), (19,3);
15/22: (67,11), (85,69);
17/22: (31,29), (23,1);
19/22: (213,53), (222,86);
21/22: (104,76), (97,31);
1/23: (16,14), (29,17);
2/23: (247,9), (458,186);
3/23: (131,73), (219,127);
4/23: (671,463), (1089,843);
5/23: (27,23), (33,13);
6/23: (88,76), (103,57);
7/23: (23,2), (31,8);
8/23: (83,77), (82,56);
9/23: (7,2), (12,11);
10/23: (107,21), (132,40);
11/23: (453,427), (369,1);
12/23: (41,23), (47,1);
13/23: (33,19), (37,9);
14/23: (274,202), (287,127);
15/23: (19,7), (25,21);
16/23: (32,28), (29,17);
17/23: (138,134), (87,41);
18/23: (161,73), (170,60);
19/23: (51,13), (56,36);
20/23: (46,22), (47,1);
21/23: (89,19), (91,1);
22/23: (181,27), (183,1);
1/24: (39,31), (76,8);
5/24: (98,64), (159,129);
7/24: (81,73), (86,46);
11/24: (23,1), (28,8);
13/24: (37,3), (62,58);
17/24: (21,13), (22,2);
19/24: (29,9), (49,47);
23/24: (24,22), (19,13);
1/25: (80,68), (149,43);
2/25: (91,83), (128,46);
3/25: (19,17), (25,5);
4/25: (34,18), (53,21);
6/25: (57,39), (89,73);
7/25: (8,1), (11,2);
8/25: (156,8), (209,87);
9/25: (9,6), (11,2);
11/25: (13,9), (15,5);
12/25: (31,17), (40,30);
13/25: (155,83), (179,53);
14/25: (31,17), (35,5);
16/25: (160,136), (149,43);
17/25: (23,11), (25,5);
18/25: (177,129), (191,137);
19/25: (61,23), (65,5);
21/25: (2894,1208), (3255,2365);
22/25: (233,31), (280,230);
23/25: (49,43), (40,10);
24/25: (172,140), (157,99);
1/26: (32,2), (77,53);
3/26: (31,29), (37,3);
5/26: (8,6), (11,3);
7/26: (12,2), (17,9);
9/26: (31,23), (37,11);
11/26: (14,8), (17,7);
15/26: (7,1), (9,7);
17/26: (26,8), (29,11);
19/26: (27,11), (29,3);
21/26: (16,2), (17,7);
23/26: (208,114), (223,153);
25/26: (209,41), (211,29);
1/27: (4,2), (9,3);
2/27: (13,11), (21,9);
4/27: (31,1), (51,27);
5/27: (16,13), (27,24);
7/27: (41,29), (54,24);
8/27: (82,22), (111,9);
10/27: (7,1), (9,3);
11/27: (7,4), (9,6);
13/27: (31,27), (32,22);
14/27: (71,57), (74,34);
16/27: (8,4), (9,3);
17/27: (8,2), (9,3);
19/27: (21,17), (28,26);
20/27: (196,128), (201,39);
22/27: (37,29), (42,36);
23/27: (32,14), (33,3);
25/27: (19,17), (15,3);
26/27: (17,7), (18,12);
1/28: (73,71), (104,76);
3/28: (26,8), (59,53);
5/28: (20,10), (31,17);
9/28: (27,21), (32,4);
11/28: (19,3), (24,4);
13/28: (25,5), (31,17);
15/28: (16,14), (15,1);
17/28: (21,13), (23,9);
19/28: (581,391), (627,381);
23/28: (39,7), (58,54);
25/28: (303,47), (312,80);
27/28: (33,27), (29,13);
1/29: (3,1), (7,3);
2/29: (53,19), (103,13);
3/29: (4,2), (7,3);
4/29: (27,11), (44,6);
5/29: (7,1), (13,11);
6/29: (17,7), (31,27);
7/29: (39,31), (49,9);
8/29: (8,4), (13,11);
9/29: (11,7), (18,16);
10/29: (59,19), (77,25);
11/29: (313,17), (403,181);
12/29: (27,21), (32,22);
13/29: (6,4), (7,3);
14/29: (67,53), (71,11);
15/29: (43,7), (68,62);
16/29: (6,2), (7,3);
17/29: (21,1), (24,2);
18/29: (299,61), (339,137);
19/29: (28,10), (31,5);
20/29: (152,84), (163,45);
21/29: (13,1), (18,16);
22/29: (103,73), (107,67);
23/29: (63,29), (66,8);
24/29: (158,14), (169,89);
25/29: (54,28), (55,3);
26/29: (74,22), (79,49);
27/29: (81,69), (113,109);
28/29: (171,101), (167,3);
1/30: (3,1), (7,1);
7/30: (113,111), (106,94);
11/30: (45,43), (37,11);
13/30: (6,4), (7,1);
17/30: (71,37), (87,63);
19/30: (908,414), (1009,313);
23/30: (77,15), (127,121);
29/30: (7,3), (7,1);
1/31: (9,8), (18,13);
2/31: (25,15), (48,14);
3/31: (41,37), (56,6);
4/31: (33,31), (39,23);
5/31: (64,14), (101,23);
6/31: (389,349), (497,373);
7/31: (529,451), (641,269);
8/31: (166,2), (233,47);
9/31: (67,49), (84,22);
10/31: (17,9), (32,30);
11/31: (37,29), (43,19);
12/31: (29,13), (38,24);
13/31: (83,57), (97,27);
14/31: (109,53), (132,54);
15/31: (16,13), (18,13);
16/31: (18,16), (18,13);
17/31: (92,91), (79,76);
18/31: (51,21), (58,4);
19/31: (24,14), (33,29);
20/31: (82,54), (87,25);
21/31: (67,31), (73,11);
22/31: (911,871), (639,291);
23/31: (313,221), (314,4);
24/31: (41,23), (43,19);
25/31: (251,193), (238,52);
26/31: (319,191), (398,346);
27/31: (94,68), (103,83);
28/31: (109,3), (113,51);
29/31: (175,173), (82,20);
30/31: (37,13), (63,61);
1/32: (58,17), (139,61);
3/32: (19,2), (39,31);
5/32: (17,4), (31,25);
7/32: (11,3), (18,14);
9/32: (8,1), (11,3);
11/32: (79,9), (122,102);
13/32: (73,31), (118,106);
15/32: (4,1), (5,3);
17/32: (19,9), (22,6);
19/32: (47,28), (53,29);
21/32: (17,11), (18,2);
23/32: (229,93), (247,9);
25/32: (64,23), (91,83);
27/32: (21,9), (26,22);
29/32: (103,13), (106,38);
31/32: (24,7), (25,15);
1/33: (3,2), (7,4);
2/33: (225,119), (448,190);
4/33: (9,7), (14,8);
5/33: (18,16), (25,19);
7/33: (23,19), (29,7);
8/33: (28,24), (37,29);
10/33: (37,3), (50,16);
13/33: (71,69), (53,31);
14/33: (11,3), (14,8);
16/33: (6,4), (7,4);
17/33: (7,6), (7,4);
19/33: (67,47), (73,37);
20/33: (29,3), (37,29);
23/33: (29,17), (31,13);
25/33: (47,21), (50,16);
26/33: (46,6), (49,17);
28/33: (222,194), (197,133);
29/33: (61,7), (63,3);
31/33: (163,147), (127,49);
32/33: (225,119), (224,95);
1/34: (14,8), (37,29);
3/34: (11,1), (21,13);
5/34: (3,1), (5,3);
7/34: (113,111), (86,18);
9/34: (69,27), (104,76);
11/34: (151,129), (171,101);
13/34: (27,1), (39,31);
15/34: (4,2), (5,3);
19/34: (32,6), (37,3);
21/34: (37,19), (41,7);
23/34: (39,7), (44,24);
25/34: (110,100), (97,71);
27/34: (99,57), (103,47);
29/34: (247,35), (257,25);
31/34: (64,2), (133,131);
33/34: (59,37), (57,7);
1/35: (3,1), (8,6);
2/35: (65,63), (82,54);
3/35: (2,1), (4,3);
4/35: (41,7), (71,29);
6/35: (53,43), (73,39);
8/35: (328,192), (509,387);
9/35: (59,49), (71,29);
11/35: (24,2), (33,19);
12/35: (337,319), (374,332);
13/35: (3,2), (4,3);
16/35: (3,1), (4,3);
17/35: (45,7), (54,16);
18/35: (267,243), (236,2);
19/35: (130,79), (148,71);
22/35: (24,20), (23,9);
23/35: (63,29), (81,67);
24/35: (10,2), (11,3);
26/35: (319,191), (332,32);
27/35: (15,3), (16,2);
29/35: (7,3), (8,6);
31/35: (199,173), (166,16);
32/35: (65,63), (41,27);
33/35: (89,67), (87,59);
34/35: (115,13), (116,32);
1/36: (7,3), (17,1);
5/36: (19,17), (27,21);
7/36: (16,2), (27,21);
11/36: (29,7), (39,9);
13/36: (194,112), (243,3);
17/36: (18,16), (17,1);
19/36: (344,74), (411,213);
23/36: (499,131), (558,138);
25/36: (22,4), (27,21);
29/36: (17,1), (18,6);
31/36: (7259,6763), (5310,390);
35/36: (34,8), (35,19);
1/37: (11,7), (26,8);
2/37: (9,7), (17,9);
3/37: (20,8), (39,25);
4/37: (26,2), (59,53);
5/37: (4,3), (6,1);
6/37: (187,131), (276,94);
7/37: (11,3), (17,9);
8/37: (12,8), (17,9);
9/37: (78,18), (137,119);
10/37: (37,29), (46,20);
11/37: (14,3), (21,16);
12/37: (47,1), (63,29);
13/37: (5,1), (7,5);
14/37: (141,139), (91,57);
15/37: (44,14), (55,3);
16/37: (11,7), (13,4);
17/37: (76,42), (123,113);
18/37: (27,21), (29,11);
19/37: (43,33), (47,27);
20/37: (229,181), (236,10);
21/37: (11,4), (19,18);
22/37: (28,16), (31,1);
23/37: (1129,669), (1332,962);
24/37: (83,77), (66,8);
25/37: (1091,687), (1153,117);
26/37: (11,3), (12,2);
27/37: (12,3), (13,4);
28/37: (31,17), (35,25);
29/37: (43,23), (58,52);
30/37: (38,34), (31,1);
31/37: (44,18), (47,27);
32/37: (18,14), (17,9);
33/37: (71,17), (73,1);
34/37: (59,53), (47,23);
35/37: (148,64), (149,43);
36/37: (499,131), (502,96);
1/38: (29,3), (72,4);
3/38: (49,47), (58,18);
5/38: (78,28), (129,23);
7/38: (274,272), (173,59);
9/38: (473,437), (499,261);
11/38: (49,39), (59,21);
13/38: (31,1), (44,32);
15/38: (112,34), (141,11);
17/38: (59,43), (67,29);
21/38: (44,26), (61,53);
23/38: (101,37), (158,146);
25/38: (25,15), (27,11);
27/38: (33,27), (31,7);
29/38: (71,11), (77,37);
31/38: (274,212), (273,183);
33/38: (923,65), (957,235);
35/38: (703,479), (682,302);
37/38: (376,142), (391,239);
1/39: (16,4), (41,23);
2/39: (9,7), (17,7);
4/39: (17,1), (32,22);
5/39: (3,1), (5,1);
7/39: (11,3), (17,7);
8/39: (12,8), (17,7);
10/39: (241,209), (281,151);
11/39: (25,19), (31,5);
14/39: (24,4), (31,1);
16/39: (32,8), (41,23);
17/39: (13,1), (16,2);
19/39: (79,3), (95,35);
20/39: (13,3), (16,10);
22/39: (67,21), (82,56);
23/39: (83,32), (97,56);
25/39: (10,5), (11,2);
28/39: (34,6), (37,11);
29/39: (62,4), (67,23);
31/39: (283,159), (292,46);
32/39: (18,14), (17,7);
34/39: (5,3), (5,1);
35/39: (37,9), (38,8);
37/39: (17,9), (17,7);
38/39: (123,67), (121,19);
1/40: (14,12), (29,3);
3/40: (31,5), (83,77);
7/40: (19,9), (29,3);
9/40: (15,9), (22,14);
11/40: (21,1), (29,3);
13/40: (11,3), (16,12);
17/40: (5,3), (6,2);
19/40: (129,23), (156,56);
21/40: (45,39), (43,11);
23/40: (133,5), (212,196);
27/40: (15,9), (16,4);
29/40: (179,53), (194,58);
31/40: (43,19), (46,22);
33/40: (163,35), (172,68);
37/40: (29,11), (38,34);
39/40: (47,25), (68,64);
1/41: (19,9), (48,22);
2/41: (17,7), (44,38);
3/41: (29,23), (51,31);
4/41: (37,11), (83,73);
5/41: (27,23), (42,32);
6/41: (14,2), (33,31);
7/41: (9,2), (14,3);
8/41: (6,2), (9,1);
9/41: (13,4), (21,16);
10/41: (26,2), (37,5);
11/41: (14,8), (19,7);
12/41: (41,23), (59,43);
13/41: (21,1), (28,6);
14/41: (57,41), (69,13);
15/41: (7,1), (9,1);
16/41: (19,9), (24,11);
17/41: (13,1), (17,11);
18/41: (37,19), (48,34);
19/41: (71,43), (83,1);
20/41: (46,22), (59,43);
21/41: (16,2), (19,7);
22/41: (71,39), (81,1);
23/41: (19,4), (22,7);
24/41: (8,4), (9,1);
25/41: (74,32), (83,1);
26/41: (9,7), (9,1);
27/41: (18,12), (19,7);
28/41: (164,24), (183,89);
29/41: (113,61), (121,43);
30/41: (197,29), (213,31);
31/41: (863,191), (925,125);
32/41: (17,7), (22,19);
33/41: (92,62), (99,71);
34/41: (129,23), (136,54);
35/41: (32,18), (43,39);
36/41: (46,26), (49,33);
37/41: (63,29), (64,18);
38/41: (847,809), (553,139);
39/41: (16,2), (17,11);
40/41: (53,21), (54,28);
1/42: (32,2), (83,43);
5/42: (43,11), (90,78);
11/42: (26,4), (37,19);
13/42: (107,63), (139,29);
17/42: (25,15), (31,17);
19/42: (318,52), (983,977);
23/42: (63,29), (73,31);
25/42: (29,3), (44,40);
29/42: (32,22), (33,9);
31/42: (79,17), (88,52);
37/42: (31,1), (32,4);
41/42: (537,487), (412,188);
1/43: (263,257), (367,23);
2/43: (77,19), (169,89);
3/43: (253,223), (449,363);
4/43: (271,33), (491,111);
5/43: (18,1), (31,12);
6/43: (529,511), (581,451);
7/43: (48,20), (75,11);
8/43: (52,16), (79,7);
9/43: (32,22), (49,37);
10/43: (33,7), (87,85);
11/43: (31,24), (39,4);
12/43: (73,71), (58,28);
13/43: (89,67), (109,23);
14/43: (391,281), (479,49);
15/43: (37,23), (50,36);
16/43: (526,514), (367,23);
17/43: (27,23), (44,42);
18/43: (409,319), (453,23);
19/43: (834,686), (879,239);
20/43: (226,218), (169,89);
21/43: (26,19), (29,14);
22/43: (59,37), (67,19);
23/43: (38,8), (49,37);
24/43: (279,87), (323,107);
25/43: (145,75), (163,9);
26/43: (755,645), (708,106);
27/43: (72,33), (152,149);
28/43: (521,151), (697,593);
29/43: (398,356), (451,409);
30/43: (91,37), (104,68);
31/43: (247,187), (257,173);
32/43: (129,1), (139,33);
33/43: (239,47), (256,84);
34/43: (117,53), (123,37);
35/43: (678,372), (701,271);
36/43: (229,203), (193,107);
37/43: (79,44), (152,149);
38/43: (381,227), (389,213);
39/43: (1216,178), (1603,1431);
40/43: (33,1), (45,41);
41/43: (44,14), (49,37);
42/43: (58,2), (59,27);
1/44: (19,13), (46,2);
3/44: (113,61), (227,147);
5/44: (42,8), (93,83);
7/44: (12,2), (19,3);
9/44: (39,9), (58,14);
13/44: (157,99), (204,28);
15/44: (146,136), (135,41);
17/44: (41,7), (52,8);
19/44: (23,15), (27,5);
21/44: (43,29), (49,17);
23/44: (539,59), (634,106);
25/44: (180,130), (259,237);
27/44: (27,21), (28,16);
29/44: (83,77), (74,58);
31/44: (226,164), (323,301);
35/44: (564,486), (521,359);
37/44: (137,119), (116,28);
39/44: (59,43), (56,12);
41/44: (57,25), (90,86);
43/44: (263,253), (169,103);
1/45: (6,4), (19,17);
2/45: (5,3), (11,7);
4/45: (9,7), (19,17);
7/45: (9,5), (14,4);
8/45: (20,16), (27,3);
11/45: (20,2), (33,27);
13/45: (8,1), (12,9);
14/45: (11,3), (19,17);
16/45: (12,8), (19,17);
17/45: (14,12), (19,17);
19/45: (20,18), (19,1);
22/45: (419,221), (492,144);
23/45: (60,14), (71,19);
26/45: (37,29), (38,16);
28/45: (101,11), (114,36);
29/45: (31,5), (42,36);
31/45: (48,14), (55,35);
32/45: (10,6), (11,7);
34/45: (55,1), (69,57);
37/45: (92,34), (99,57);
38/45: (277,163), (326,268);
41/45: (10,8), (9,3);
43/45: (876,586), (839,227);
44/45: (328,284), (603,597);
1/46: (63,59), (117,67);
3/46: (13,7), (27,19);
5/46: (106,98), (133,5);
7/46: (181,113), (311,241);
9/46: (91,37), (154,122);
11/46: (37,29), (47,1);
13/46: (43,7), (59,13);
15/46: (70,50), (87,41);
17/46: (83,77), (76,16);
19/46: (439,55), (670,578);
21/46: (139,71), (167,63);
25/46: (430,320), (457,49);
27/46: (489,183), (556,88);
29/46: (113,55), (125,13);
31/46: (43,19), (47,1);
33/46: (34,32), (27,19);
35/46: (2503,633), (2802,1790);
37/46: (433,399), (334,126);
39/46: (151,47), (157,19);
41/46: (69,47), (67,21);
43/46: (354,334), (243,13);
45/46: (694,82), (1111,1065);
1/47: (7,6), (24,23);
2/47: (36,32), (63,31);
3/47: (65,61), (89,5);
4/47: (59,43), (109,79);
5/47: (191,149), (320,226);
6/47: (263,7), (454,266);
7/47: (19,9), (48,46);
8/47: (72,32), (111,17);
9/47: (41,23), (96,92);
10/47: (1057,799), (2299,2213);
11/47: (21,1), (48,46);
12/47: (151,73), (223,153);
13/47: (57,53), (58,36);
14/47: (73,31), (98,4);
15/47: (22,7), (30,17);
16/47: (14,12), (24,23);
17/47: (29,22), (34,13);
18/47: (197,181), (209,167);
19/47: (101,89), (109,79);
20/47: (349,307), (344,62);
21/47: (148,22), (181,7);
22/47: (1969,1111), (2318,344);
23/47: (3471,3061), (3726,2948);
24/47: (724,32), (863,359);
25/47: (1671,1153), (1835,237);
26/47: (379,107), (483,363);
27/47: (51,39), (56,38);
28/47: (55,41), (59,35);
29/47: (173,117), (187,93);
30/47: (64,52), (63,31);
31/47: (232,109), (304,257);
32/47: (72,64), (63,31);
33/47: (61,28), (66,19);
34/47: (83,53), (86,8);
35/47: (268,226), (301,263);
36/47: (459,123), (514,332);
37/47: (55,3), (60,34);
38/47: (479,471), (268,174);
39/47: (83,77), (63,31);
40/47: (29,3), (48,46);
41/47: (327,81), (371,277);
42/47: (2295,321), (2560,1858);
43/47: (971,799), (908,626);
44/47: (76,56), (71,23);
45/47: (623,389), (632,402);
46/47: (11651,10613), (8753,1493);
1/48: (3,1), (8,4);
5/48: (3,2), (5,1);
7/48: (13,1), (22,14);
11/48: (9,2), (13,1);
13/48: (3,2), (4,2);
17/48: (21,1), (28,16);
19/48: (13,6), (17,11);
23/48: (219,127), (262,146);
25/48: (25,5), (38,34);
29/48: (7,3), (8,4);
31/48: (28,3), (41,37);
35/48: (4,3), (4,2);
37/48: (39,25), (40,16);
41/48: (51,31), (58,46);
43/48: (244,57), (467,457);
47/48: (89,5), (130,122);
1/49: (20,17), (44,5);
2/49: (17,9), (51,47);
3/49: (311,301), (379,209);
4/49: (36,32), (57,41);
5/49: (51,49), (59,39);
6/49: (113,7), (194,96);
8/49: (102,94), (119,63);
9/49: (303,81), (482,302);
10/49: (203,53), (334,254);
11/49: (217,173), (277,17);
12/49: (214,178), (259,77);
13/49: (197,133), (316,272);
15/49: (39,36), (38,11);
16/49: (40,34), (44,5);
17/49: (28,23), (32,17);
18/49: (29,11), (51,47);
19/49: (91,61), (109,11);
20/49: (289,273), (243,47);
22/49: (61,27), (74,24);
23/49: (133,87), (153,43);
24/49: (31,1), (51,47);
25/49: (149,43), (176,20);
26/49: (401,367), (422,362);
27/49: (83,79), (64,34);
29/49: (46,12), (99,97);
30/49: (509,187), (573,113);
31/49: (142,44), (161,77);
32/49: (19,11), (26,23);
33/49: (238,4), (263,67);
34/49: (277,243), (274,216);
36/49: (113,49), (121,23);
37/49: (51,47), (42,28);
38/49: (163,11), (181,113);
39/49: (64,34), (77,63);
40/49: (129,31), (189,175);
41/49: (761,121), (854,602);
43/49: (88,2), (107,89);
44/49: (469,139), (481,89);
45/49: (849,531), (832,50);
46/49: (3903,959), (4008,1852);
47/49: (118,117), (51,2);
48/49: (384,204), (581,553);
1/50: (133,131), (175,65);
3/50: (86,70), (157,99);
7/50: (397,373), (463,287);
9/50: (26,8), (69,67);
11/50: (217,135), (308,144);
13/50: (5,1), (7,1);
17/50: (66,26), (87,41);
19/50: (631,433), (755,125);
21/50: (114,72), (139,77);
23/50: (33,13), (69,67);
27/50: (89,73), (95,65);
29/50: (267,43), (306,58);
31/50: (1234,1184), (875,355);
33/50: (53,31), (57,1);
37/50: (7,5), (7,1);
39/50: (23,11), (25,15);
41/50: (53,29), (55,25);
43/50: (401,143), (415,95);
47/50: (688,534), (755,645);
49/50: (151,53), (173,139);
Aquí está la lista de los más pequeños de soluciones que han$\max\{a,c\}\ge 10000$$q\le 130$:
46/47: (11651,10613), (8753,1493);
30/67: (8267,2821), (10151,4255);
43/67: (111359,71993), (118701,29637);
67/68: (12813,8659), (12250,5430);
67/70: (11309,7853), (10702,1198);
43/71: (15847,13909), (14344,854);
67/71: (21498,18568), (20729,17037);
62/79: (10063,2127), (15066,14006);
79/83: (53629,23159), (53817,1029);
86/93: (5371,2189), (10242,10032);
86/97: (14552,9528), (14379,6187);
59/98: (14107,301), (17921,13903);
21/100: (8381,5717), (12935,9895);
53/103: (9281,1611), (10956,1198);
60/103: (11219,1789), (13480,8742);
47/107: (18727,12523), (28660,25910);
61/107: (21361,13787), (24487,15499);
86/107: (9453,4651), (10983,8491);
103/107: (12381,10897), (9941,1401);
106/107: (39212,8912), (39281,5231);
18/121: (20721,17817), (28028,15334);
46/121: (13881,8383), (17061,2761);
67/121: (14463,7227), (20893,18473);
84/121: (12982,1054), (23089,22209);
103/122: (24576,12916), (27917,21363);
43/124: (9281,6271), (12234,8598);
75/124: (16829,16579), (12634,11546);
79/124: (49279,33039), (52132,816);
121/124: (17452,12128), (16433,2353);
35/127: (8637,4059), (12403,8171);
52/127: (7275,2053), (10680,8878);
71/127: (20579,11917), (23843,14003);
84/127: (13445,1675), (17768,14974);
116/127: (29409,16353), (29516,11632);
118/127: (9435,4125), (13313,12341);
123/127: (16159,1717), (16318,4764);
124/129: (9578,4682), (10729,8407);
59/130: (8432,1126), (10275,1845);
Y algunos de los más difíciles ejemplos para$p=163$$q=163$:
38/163: (10722,10026), (11191,6953);
56/163: (29626,23154), (34641,13607);
79/163: (14697,12643), (17178,14444);
82/163: (???,???), (???,???)
114/163: (28501,17899), (30299,14651);
121/163: (???,???), (???,???)
142/163: (11646,7382), (14751,13121);
150/163: (53691,44613), (47312,23104);
151/163: (30839,17783), (30538,5974);
155/163: (13770,11650), (11653,1221);
163/166: (???,???), (???,???)
163/167: (???,???), (???,???)
163/178: (20459,12309), (31496,30072);
163/181: (16136,12226), (14989,1663);
163/190: (???,???), (???,???)
163/191: (???,???), (???,???)
163/193: (18647,2543), (19451,2467);
163/196: (11081,10109), (12081,11199);
163/199: (???,???), (???,???)
163/206: (19151,7907), (20201,6193);
