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Let $ S = 1! ~2!~\dotsm ~100! $. Prove that there exists a unique positive integer $k$ such that $S/k!$ is a perfect square.

I thought this was a cute, fun problem and I did solve it, but any alternative methods that you guys would use?

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1 Answer 1

EDIT, Friday morning; testing with changing the 100 to other numbers. It appears that if, as above, $S = 1! \cdot 2! \cdot 3! \cdot \cdots \cdot (4n)!,$ then $S / (2n)!$ is a square.

EEDDIITT TTOOOOO: need the larger number divisible by 4, this thing fails when replacing 100 by 6 or 10. INDUCTION: Given that $$ T = 1! \cdot 2! \cdot 3! \cdot \cdots \cdot (4n - 4)!, $$ and $$ T / (2n-2)! = \; \; \mbox{square}. $$ Let $$ S = 1! \cdot 2! \cdot 3! \cdot \cdots \cdot (4n)!. $$ Then $$ \color{magenta}{ S / (2n)! = \left( T / (2n-2)! \right) \cdot ((4n-4)!)^4 \cdot (4n-3)^4 \cdot (4n-2)^2 \cdot (4n-1)^2 \cdot 4}$$ is still a square. I do not immediately see uniqueness in adequate detail, but we know we cannot perturb the $2n$ as high as the next prime, or lower than the previous prime. Maybe some way to fill in an argument. Or, uniqueness in general could be false, occasionally more than one answer when prime gaps are unusually large. Not sure.

ORIGINAL: Well, it's 50. Because of prime 101, $k$ cannot exceed 100. Because of all the even exponents, actually $k$ cannot exceed 52. Because of the odd exponents just below, $k$ must be at least 47. In brief, because of prime 2, $k$ is at least 50. Because of prime 17, $k$ is at most 50. After that it is a matter of matching in detail and checking that 50 actually works for each prime from 2 to 47, which it does.

Note $$ 50! = 2^{47} \cdot 3^{22} \cdot 5^{12} \cdot 7^8 \cdot 11^4 \cdot 13^3 \cdot 17^2 \cdot 19^2 \cdot 23^2 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \cdot 47. $$

$S:$

    2        4731    1
    3        2328    0
    5        1124    0
    7         734    0
   11         414    0
   13         343    1
   17         250    0
   19         220    0
   23         174    0
   29         129    1
   31         117    1
   37          91    1
   41          79    1
   43          73    1
   47          61    1
   53          48    0
   59          42    0
   61          40    0
   67          34    0
   71          30    0
   73          28    0
   79          22    0
   83          18    0
   89          12    0
   97           4    0

   47           2   42
   48           2   46
   49           2   46
   50           2   47
   51           2   47
   52           2   49

   47           3   21
   48           3   22
   49           3   22
   50           3   22
   51           3   23
   52           3   23

   47           5   10
   48           5   10
   49           5   10
   50           5   12
   51           5   12
   52           5   12

   47           7    6
   48           7    6
   49           7    8
   50           7    8
   51           7    8
   52           7    8

   47          11    4
   48          11    4
   49          11    4
   50          11    4
   51          11    4
   52          11    4

   47          13    3
   48          13    3
   49          13    3
   50          13    3
   51          13    3
   52          13    4

   47          17    2
   48          17    2
   49          17    2
   50          17    2
   51          17    3
   52          17    3

   47          19    2
   48          19    2
   49          19    2
   50          19    2
   51          19    2
   52          19    2

   47          23    2
   48          23    2
   49          23    2
   50          23    2
   51          23    2
   52          23    2

   47          29    1
   48          29    1
   49          29    1
   50          29    1
   51          29    1
   52          29    1

   47          31    1
   48          31    1
   49          31    1
   50          31    1
   51          31    1
   52          31    1

   47          37    1
   48          37    1
   49          37    1
   50          37    1
   51          37    1
   52          37    1

   47          41    1
   48          41    1
   49          41    1
   50          41    1
   51          41    1
   52          41    1

   47          43    1
   48          43    1
   49          43    1
   50          43    1
   51          43    1
   52          43    1

   47          47    1
   48          47    1
   49          47    1
   50          47    1
   51          47    1
   52          47    1

=================

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