Factorial is fun!

Number Theory Level 5

3 2 = 1 ! + 2 ! + 3 ! 1 2 2 = 4 ! + 5 ! 7 1 2 = 1 ! + 7 ! \begin{aligned} 3^2 &= 1!+2!+3! \\ 12^2&= 4!+5! \\ 71^2 &= 1!+7! \\ \end{aligned}

How many square numbers less than 21 ! 21! can be expressed as a sum of distinct factorials (including the three above)?


The answer is 15.

Harsh Shrivastava by Harsh Shrivastava , 17, India

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1 solution

Michael Mendrin
Jul 22, 2018

1 2 = 1 ! 1^2= 1!
3 2 = 1 ! + 2 ! + 3 ! 3^2= 1! + 2! + 3!
5 2 = 1 ! + 4 ! 5^2= 1! + 4!
1 1 2 = 1 ! + 5 ! 11^2= 1! + 5!
1 2 2 = 4 ! + 5 ! 12^2= 4! + 5!
2 7 2 = 1 ! + 2 ! + 3 ! + 6 ! 27^2= 1! + 2! + 3!+6!
2 9 2 = 1 ! + 5 ! + 6 ! 29^2= 1! + 5! + 6!
7 1 2 = 1 ! + 7 ! 71^2= 1! + 7!
7 2 2 = 4 ! + 5 ! + 7 ! 72^2= 4! + 5! + 7!
60 3 2 = 1 ! + 2 ! + 3 ! + 6 ! + 9 ! 603^2=1! + 2! + 3! + 6! + 9!
63 5 2 = 1 ! + 4 ! + 8 ! + 9 ! 635^2=1! + 4! + 8! + 9!
21 3 2 = 1 ! + 2 ! + 3 ! + 7 ! + 8 ! 213^2=1! + 2! + 3! +7!+ 8!
21 5 2 = 1 ! + 4 ! + 5 ! + 6 ! + 7 ! + 8 ! 215^2=1! + 4! + 5!+6! +7!+ 8!
191 7 2 = 1 ! + 2 ! + 3 ! + 6 ! + 7 ! + 8 ! + 10 ! 1917^2=1! + 2! + 3! +6!+ 7! + 8! + 10!
118389 3 2 = 1 ! + 2 ! + 3 ! + 7 ! + 8 ! + 9 ! + 10 ! + 11 ! + 12 ! + 13 ! + 14 ! + 15 ! 1183893^2=1! + 2! + 3! + 7 ! + 8! + 9! + 10! + 11! + 12! + 13! + 14! + 15!


4 Helpful 0 Interesting 0 Brilliant 0 Confused

Did you calculate all of this... How did u reach here????

Samarth Hegde - 2 years, 10 months ago

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I was only able to do this with a computer search, and even then it took time. The next such square is likely to be very large. This problem should be "Computer Science".

Michael Mendrin - 2 years, 10 months ago

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According to OEIS the next largest such square, if there is one, would be > 48 ! \gt 48! .

Brian Charlesworth - 2 years, 10 months ago

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@Brian Charlesworth – I just now checked OEIS. Wow, what if the author of this problem had asked for how many squares < 48 ! <48! ?

Michael Mendrin - 2 years, 10 months ago

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@Michael Mendrin – Haha Then your program to find the squares would still be running. :)

Brian Charlesworth - 2 years, 10 months ago

Did you calculate all of this... How did u reach here????

Daanish bansal - 2 years, 5 months ago

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