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Friend gave me this problem (although I don't think he knows the answer). The question is whether the product $$ \frac{3m_1+2}{2m_1+1}\cdot\frac{3m_2+2}{2m_2+1}\cdots \frac{3m_n+2}{2m_n+1} $$ can be an integer for some distinct $m_i \in \mathbb{N}$ (here $0 \not\in \mathbb{N}$, otherwise we would have simple solution $n=1, m_1=0$, giving product $2$). So basically this is about products of combinations of numbers $$ \frac{5}{3},\frac{8}{5},\frac{11}{7},\frac{14}{9}, \frac{17}{11}, \frac{20}{13}, \frac{23}{15}, \frac{26}{17}, \frac{29}{19}, \frac{32}{21}, \dots $$

My try so far:

We can rule out fractions which have denominator divisible by $3$, since no numerator can be divisible by $3$, so basically $m \not\equiv 1 \pmod 3$. I have then tried computer checking all products with $m_i \in \{2,3,5,6,8,9,\dots,39\}$, none of them yield an integer.

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6
  • 2
    $\begingroup$ Preliminary ideas: It's trivial to see that no product of two $m_k$'s is an integer, and some deduction shows that no product of three $m_k$'s is an integer, either. $\endgroup$ Jun 9, 2018 at 8:34
  • 2
    $\begingroup$ No product of four $m_k$'s, either. $\endgroup$ Jun 9, 2018 at 8:41
  • $\begingroup$ I wrote a small program that tests whether the value is an integer and the first result of this occurring when $m_1=2$ and $m_i=m_{i-1}+1$ is when $m_n=82$ giving a result of $338480466710618$. I suspect you want a more mathematically based answer, however. $\endgroup$ Jun 9, 2018 at 8:50
  • $\begingroup$ @CooperCape That would be alright, but I don't see how it works, since for example $m_3=4$ and so $2m_3+1=9$ is divisible by $3$, and there can be no corresponding numerator in which this would cancel (none of them is divisible by $3$). Are you sure your result is correct? $\endgroup$
    – Sil
    Jun 9, 2018 at 8:57
  • $\begingroup$ Okay, give me one sec, I'll check it numerically. $\endgroup$ Jun 9, 2018 at 8:59

1 Answer 1

7
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There are solutions for many values of $n$. You can find here solutions for $n=5,\;\;7,8,9,10,\ldots, 21$.

One of solutions for $n=5$:

$$ (m_1,m_2,m_3,m_4,m_5)=(9,12,14,27,41): $$

$$ \dfrac{3\cdot 9+2}{2\cdot 9 + 1} \times \dfrac{3\cdot 12+2}{2\cdot 12 + 1} \times \dfrac{3\cdot 14+2}{2\cdot 14 + 1} \times \dfrac{3\cdot 27+2}{2\cdot 27 + 1} \times \dfrac{3\cdot 41+2}{2\cdot 41 + 1} = \\ \dfrac{\color{red}{29}}{\color{violet}{19}} \times \dfrac{\color{violet}{38}}{\color{blue}{25}} \times \dfrac{44}{\color{red}{29}} \times \dfrac{\color{green}{83}}{55} \times \dfrac{\color{blue}{125}}{\color{green}{83}} = \\ \dfrac{503063000}{62882875} = 8.$$

Another examples of solutions for $n=5$:
$(9,12,14,21,71) \rightarrow 8$;
$(6,12,27,41,47) \rightarrow 8$;
$(6,12,21,47,71) \rightarrow 8$;
$(9, 11, 12, 30, 101) \rightarrow 8$;
$(8, 9, 14, 71, 107) \rightarrow 8$;
$(6, 8, 47, 71, 107) \rightarrow 8$;
$(5, 12, 41, 47, 110) \rightarrow 8$;
$(6, 8, 39, 87, 131) \rightarrow 8$;
$(6, 11, 17, 99, 132) \rightarrow 8$;
$(3, 99, 123, 126, 132) \rightarrow 8$;
$\ldots$
$(3,30,1943,5351,5652) \rightarrow 8$;
$(3,30,2081,4371,5912) \rightarrow 8$;
$(3,30,2133,4680,5103) \rightarrow 8$;
$\ldots$


Now focus on $n\ge 7$. For each $n\ge 7$ we search integer solution $q, (m_1,m_2,\ldots,m_n)$ for equation $$ \prod_{j=1}^n \dfrac{3m_j+2}{2m_j+1} = q \tag{*}. $$

For each $n\ge 7$ there are many solutions of eq. $(*)$. Few examples with rather small values $m_j$:

  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)

Note 1: if $n\ge 8$, then eq. $(*)$ has solutions for different integer $q$.

Note 2: this integer product is divisible by $2$, $4$, $8$ in many cases (depends on parity of numbers $m_j$); solutions with odd $q$ exist too, but are much more rare.

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9
  • $\begingroup$ Nice! How did you find those? Looks strange that all of those are $8$ :o $\endgroup$
    – Sil
    Jun 9, 2018 at 9:22
  • 5
    $\begingroup$ @sil: Each fraction is slightly bigger than $\frac{3}{2}$, and $(3/2)^5 \approx 7.6$. A product of five such fractions is going to be a little larger than $7.6$, so you'd expect most examples with five terms to have a product of $8$. $\endgroup$
    – user14972
    Jun 9, 2018 at 9:36
  • 2
    $\begingroup$ @Oleg567 $n≠6,9,12$ since all numerators are not divisible by $3$. $\endgroup$ Jun 9, 2018 at 9:36
  • 1
    $\begingroup$ @AlexFrancisco Could you elaborate? I don't see why $\Pi_{k=1}^n 2m_k + 1$ would be a multiple of $3$ if $n=6$ (for example if all $m_k$ are $0 \mod 3$ the product will be $1 \mod 3$). Or is there another reason than the denominator being a multiple of $3$? $\endgroup$
    – ploosu2
    Jun 9, 2018 at 16:36
  • 1
    $\begingroup$ @Sil: if you are interested in $n>5$ then you can see few solutions for $n\ge 8$ in updated answer. (And thanks to you and to your friend for interesting problem). $\endgroup$
    – Oleg567
    Jun 10, 2018 at 15:18

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