Hay soluciones para muchos de los valores de $n$.
Usted puede encontrar aquí las soluciones para $n=5,\;\;7,8,9,10,\ldots, 21$.
Una de las soluciones para $n=5$:
$$
(m_1,m_2,m_3,m_4,m_5)=(9,12,14,27,41):
$$
$$
\dfrac{3\cdot 9+2}{2\cdot 9 + 1} \times
\dfrac{3\cdot 12+2}{2\cdot 12 + 1} \times
\dfrac{3\cdot 14+2}{2\cdot 14 + 1} \times
\dfrac{3\cdot 27+2}{2\cdot 27 + 1} \times
\dfrac{3\cdot 41+2}{2\cdot 41 + 1} = \\
\dfrac{\color{red}{29}}{\color{violeta}{19}} \times \dfrac{\color{violeta}{38}}{\color{blue}{25}} \times \dfrac{44}{\color{red}{29}}
\times \dfrac{\color{verde}{83}}{55} \times \dfrac{\color{blue}{125}}{\color{verde}{83}} =
\\ \dfrac{503063000}{62882875} =
8.$$
Another examples of solutions for $n=5$:
$(9,12,14,21,71) \rightarrow 8$;
$(6,12,27,41,47) \rightarrow 8$;
$(6,12,21,47,71) \rightarrow 8$;
$(9, 11, 12, 30, 101) \rightarrow 8$;
$(8, 9, 14, 71, 107) \rightarrow 8$;
$(6, 8, 47, 71, 107) \rightarrow 8$;
$(5, 12, 41, 47, 110) \rightarrow 8$;
$(6, 8, 39, 87, 131) \rightarrow 8$;
$(6, 11, 17, 99, 132) \rightarrow 8$;
$(3, 99, 123, 126, 132) \rightarrow 8$;
$\ldots$
$(3,30,1943,5351,5652) \rightarrow 8$;
$(3,30,2081,4371,5912) \rightarrow 8$;
$(3,30,2133,4680,5103) \rightarrow 8$;
$\ldots$
Ahora se centran en la $n\ge 7$.
Para cada una de las $n\ge 7$ buscamos entero solución de $q, (m_1,m_2,\ldots,m_n)$ para la ecuación
$$
\prod_{j=1}^n \dfrac{3m_j+2}{2m_j+1} = p \etiqueta{*}.
$$
For each $n\ge 7$ there are many solutions of eq. $(*)$.
Algunos ejemplos con los más pequeños los valores de $m_j$:
n q (m_1, m_2, ..., m_n)
7 19 (3,5,11,23,35,53,297)
7 19 (3,5,9,23,35,297,1551)
7 19 (3,5,11,17,53,59,4743)
... ... ...
8 26 (21,71,147,221,237,255,267,356)
8 28 (3,8,27,32,41,47,71,107)
... ... ...
9 40 (12,27,41,45,47,51,68,81,108)
9 41 (3,21,47,71,137,147,221,227,341)
9 43 (3,5,15,17,23,35,53,501,609)
9 44 (3,5,8,11,12,27,41,201,237)
... ... ...
10 58 (57,123,605,671,753,821,1004,1079,1145,1227)
10 59 (21,27,47,71,207,297,305,311,327,491)
10 61 (5,11,47,69,71,107,161,339,467,509)
10 62 (3,27,29,41,47,71,107,137,144,311)
10 64 (6,8,9,12,14,27,41,47,71,107)
10 65 (3,5,11,15,23,35,53,57,605,671)
... ... ...
11 88 (47,71,81,87,107,108,131,144,197,296,311)
11 89 (21,27,41,47,71,137,147,201,221,237,563)
11 91 (5,21,23,71,177,201,297,389,417,447,671)
11 92 (8,15,23,35,39,45,51,53,68,87,131)
11 94 (5,8,11,17,47,71,107,201,237,291,297)
11 95 (3,5,21,47,71,147,221,297,305,327,491)
11 98 (3,5,8,11,21,71,137,147,201,221,237)
11 100 (3,5,8,11,15,20,23,35,53,227,341)
11 104 (2,3,5,11,15,23,35,53,59,447,671)
... ... ...
12 131 (47,107,123,291,297,311,467,563,579,587,963,1091)
12 134 (9,21,71,164,227,297,333,341,389,417,447,513)
12 136 (11,15,23,35,53,87,131,144,192,197,296,311)
12 139 (3,21,39,71,87,131,137,147,161,221,339,509)
12 140 (9,11,12,15,20,23,101,177,201,237,291,335)
12 142 (3,11,15,23,35,53,72,357,381,795,801,1017)
12 143 (3,11,15,23,27,35,53,87,131,207,311,671)
12 145 (3,5,9,23,35,53,297,333,351,357,527,791)
12 146 (3,5,11,15,23,35,53,237,501,563,845,924)
12 148 (3,5,8,15,23,32,35,51,53,171,333,357)
12 160 (2,3,5,8,11,15,20,23,35,53,227,341)
... ... ...
13 199 (21,39,71,87,131,137,147,201,221,465,671,1079,1127)
13 200 (21,27,41,71,87,131,197,212,296,311,333,357,536)
13 202 (8,27,41,71,107,201,237,287,351,431,437,527,791)
13 203 (9,23,35,47,53,71,107,297,333,351,357,527,791)
13 205 (9,11,23,35,53,101,107,297,351,527,579,587,791)
13 206 (5,21,27,41,47,71,201,237,287,297,431,869,1304)
13 208 (6,8,21,39,71,87,131,147,221,237,255,267,356)
13 211 (3,11,27,39,41,87,131,237,263,395,593,753,773)
13 212 (9,11,12,15,20,21,23,71,101,177,201,237,335)
13 214 (3,5,21,47,71,87,131,297,305,671,851,1131,1212)
13 215 (3,5,21,27,41,137,147,201,221,237,501,609,759)
13 217 (3,5,11,23,35,53,227,297,341,389,417,447,671)
13 220 (3,5,11,15,23,35,53,87,131,144,197,296,311)
13 224 (3,5,8,11,12,27,41,99,123,132,149,168,224)
... ... ...
14 295 (51,137,147,171,201,221,389,417,423,447,671,821,971,1317)
14 296 (51,77,116,137,147,171,212,221,275,311,413,620,731,1097)
14 302 (9,21,71,107,159,201,237,239,359,563,620,705,845,855)
14 304 (8,27,41,71,107,137,147,201,206,221,237,255,267,356)
14 305 (9,17,21,71,137,147,201,221,237,291,297,333,345,1151)
14 308 (8,21,27,39,41,71,80,87,131,137,147,201,221,237)
14 310 (9,12,21,27,41,47,71,101,107,201,237,335,501,609)
14 311 (3,21,71,137,147,201,221,237,287,291,431,647,971,1347)
14 316 (3,8,39,87,131,137,147,221,227,341,389,416,417,447)
14 320 (6,11,12,15,23,35,47,51,53,71,107,161,171,242)
14 322 (3,11,15,23,35,51,53,60,87,131,171,179,513,837)
14 323 (3,11,15,23,27,35,41,53,137,147,221,237,311,563)
14 328 (3,9,11,12,27,30,41,47,101,137,147,221,227,341)
14 332 (3,5,9,12,27,41,47,71,107,144,159,201,239,359)
14 344 (2,3,11,15,23,27,35,41,53,107,237,501,609,759)
14 352 (2,3,5,11,15,23,35,53,87,131,144,197,296,311)
... ... ...
15 448 (21,71,77,80,116,137,147,206,221,237,255,267,356,357,536)
15 452 (8,71,87,107,131,197,201,237,291,296,333,357,489,759,795)
15 455 (9,21,47,71,101,137,147,201,221,335,423,471,707,927,1061)
15 458 (8,27,39,41,71,87,107,131,201,203,237,333,357,381,399)
15 460 (9,12,21,71,101,137,147,201,221,237,291,333,335,444,465)
15 464 (8,9,27,41,71,80,107,147,221,237,255,267,356,396,467)
15 470 (9,11,15,17,23,35,53,237,291,297,311,333,357,536,563)
15 472 (8,11,15,23,35,45,51,53,68,71,107,212,255,275,413)
15 476 (3,11,27,41,47,71,107,147,221,237,255,267,356,357,536)
15 481 (3,11,15,23,35,51,53,137,171,333,357,383,575,863,1295)
15 484 (3,11,15,23,27,35,41,53,164,237,287,344,431,759,795)
15 485 (3,11,15,17,23,35,53,71,107,237,291,333,705,857,1131)
15 488 (3,9,12,21,27,32,41,71,87,131,197,296,311,333,467)
15 490 (3,5,8,21,71,137,147,201,221,227,237,341,471,707,1061)
15 496 (3,5,8,27,32,41,47,63,84,95,143,164,215,323,485)
15 508 (3,5,8,11,15,20,23,35,53,71,107,465,1079,1145,1227)
15 512 (3,6,9,11,12,14,21,27,41,62,71,99,117,132,176)
... ... ...
16 676 (21,27,41,71,80,137,147,221,237,255,267,356,383,575,863,1295)
16 680 (9,21,71,137,147,201,221,237,333,395,489,563,591,593,674,753)
16 682 (9,15,51,107,171,305,327,333,467,491,501,579,587,753,795,1004)
16 686 (8,21,39,71,87,131,137,147,201,221,237,287,291,431,759,795)
16 688 (8,27,39,41,71,80,87,131,147,221,237,255,267,356,396,467)
16 692 (8,9,39,71,87,107,131,201,237,291,333,335,467,521,1092,1095)
16 700 (9,15,23,27,35,41,51,53,80,137,147,171,221,333,396,467)
16 704 (9,12,23,27,35,41,47,53,71,80,107,110,203,207,246,311)
16 710 (9,11,12,15,23,35,51,53,171,227,341,389,417,447,449,671)
16 712 (9,11,12,15,23,35,51,53,87,131,171,207,276,333,375,500)
16 716 (3,23,27,35,41,47,53,57,71,86,107,227,297,341,389,417)
16 728 (3,11,15,17,23,35,53,71,107,237,291,333,444,465,620,632)
16 736 (3,5,8,45,51,68,77,116,135,137,180,240,383,575,863,1295)
16 742 (3,5,8,21,27,41,71,137,147,201,221,237,287,431,647,971)
16 752 (3,5,8,15,23,32,35,51,53,57,171,333,357,536,605,908)
16 760 (3,5,8,11,15,23,32,35,53,87,131,171,669,671,717,1076)
... ... ...
17 1024 (8,47,71,77,107,116,141,144,188,192,212,216,227,288,341,384,512)
17 1028 (9,21,35,53,71,87,131,197,296,311,333,357,449,609,812,1091,1113)
17 1034 (8,27,39,41,47,71,87,107,131,147,201,221,467,579,609,759,795)
17 1036 (9,21,23,35,53,71,80,137,147,221,275,297,381,413,617,620,801)
17 1040 (8,15,39,45,51,68,81,87,108,131,164,227,341,389,417,447,671)
17 1048 (9,15,23,27,35,41,53,80,101,137,147,221,237,396,563,567,756)
17 1052 (6,11,23,35,47,53,71,107,171,297,389,417,447,669,671,759,1139)
17 1054 (9,11,12,21,27,71,101,201,207,237,311,335,437,501,609,881,1161)
17 1060 (9,12,15,21,23,35,51,53,71,171,287,333,431,435,513,653,684)
17 1064 (9,11,12,21,23,35,53,71,87,131,197,296,297,311,333,375,500)
17 1072 (3,23,27,35,41,47,53,71,107,144,192,201,203,246,287,431,647)
17 1088 (3,15,23,27,35,41,47,53,57,71,86,107,110,144,192,207,311)
17 1096 (3,9,12,21,27,41,53,71,80,159,239,249,332,359,539,809,867)
17 1100 (3,11,12,15,23,35,51,53,60,171,201,237,287,291,431,647,971)
17 1136 (3,5,8,11,15,23,35,53,71,107,237,333,357,416,462,867,1301)
... ... ...
18 1544 (8,21,39,71,87,131,147,201,221,237,255,267,356,396,423,467,471,707)
18 1552 (8,11,71,99,107,123,132,137,147,171,201,221,237,333,566,705,857,1131)
18 1568 (8,11,21,39,71,87,131,137,147,201,206,221,237,255,267,356,375,500)
18 1580 (8,15,23,27,35,41,51,53,80,137,147,171,221,227,341,389,417,447)
18 1592 (6,8,27,39,41,47,71,87,107,131,137,147,201,221,465,671,1079,1127)
18 1600 (9,12,15,23,27,35,41,47,53,71,101,107,126,212,237,333,396,416)
18 1616 (9,11,12,15,21,23,35,53,71,80,101,137,147,221,237,396,416,437)
18 1664 (6,8,9,12,14,21,27,39,41,71,87,131,147,221,237,255,267,356)
18 1696 (3,5,11,12,15,23,35,51,53,171,237,263,282,291,333,357,647,971)
... ... ...
19 2336 (6,15,41,47,137,147,164,221,227,237,276,341,389,417,447,501,563,845,924)
19 2366 (9,15,23,27,35,41,51,53,80,137,147,171,221,237,263,333,395,593,753)
19 2368 (6,8,39,51,77,87,116,131,137,147,171,212,221,275,311,413,620,731,1097)
19 2380 (8,11,15,23,35,51,53,71,107,171,179,237,287,431,759,795,821,837,1116)
19 2384 (8,11,12,27,41,47,71,87,107,131,197,201,237,291,296,311,335,467,1092)
19 2392 (9,12,15,21,23,35,53,71,80,81,101,108,237,335,423,471,707,927,1061)
19 2432 (9,12,14,15,23,27,35,41,45,51,53,68,83,125,222,333,357,536,563)
19 2560 (3,5,8,11,15,23,35,53,87,131,144,192,197,296,311,396,467,977,1466)
... ... ...
20 3500 (9,21,23,35,53,71,80,137,147,221,227,297,341,389,417,447,671,821,971)
20 3520 (8,15,39,45,51,68,81,87,108,117,131,176,185,237,278,287,291,431,759,795)
20 3584 (6,11,24,27,41,47,69,92,99,123,126,132,147,221,237,255,267,356,357,536)
... ... ...
21 5488 (6,11,15,17,21,23,35,53,71,80,99,132,137,147,171,221,237,471,669,707,1061)
21 5504 (9,12,14,15,23,27,35,41,45,51,53,57,68,80,86,137,147,221,237,291,759)
Nota 1: si $n\ge 8$, luego de eq. $(*)$ tiene soluciones para los diferentes entero $q$.
Nota 2: este producto entero es divisible por $2$, $4$, $8$ en muchos casos
(depende de la paridad de los números de $m_j$); soluciones con extraña $q$ existen también, pero es mucho más raro.
