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1 = 1^2

9 = 3^2

1 = 1^2

81 = 9^2

729 = 27^2

225 = 15^2

324 = 18^2

X

82944 = 288^2

176400 = 420^2

215296 = 464^2

3444736 = 1856^2

So, I am trying to find short method to find factorials. In order to achieve this, I imagined factorials as squares, one edge of which corresponding square root of said factorial. However, as these square roots tend not to be natural numbers but have decimal extension, I chose the next bigger number mimicking factorial edge. That slightly bigger edge I then squared and from it I subtracted the real factorial value in a hope to find a series of natural numbers of some sense...And actually at first I was quite delighted while finding series presented above corresponding factorials 4!, 5!, 6!, 7!, 8!, 9!, 10!, 11!, 13!, 14!, 15!, 16!

As an example, please, look at number 225 (=15^2) in the list above. It associates with 10! (=3628800) in the following way:

10! squared (=10^(1/2)) = 1904,940944...

So, I chose 1905 as enlarged edge of square: 1905^2 = 3629025

From that I subtracted the real value of 10!: 3629025 - 3628800 = 225 = 15^2.

In an analogous way I found other squares presented in the list...but not that one corresponding 12! as 12!^(1/2) = 21886,10518... and 21887^2 - 12! = 39169 is not a square of any natural number. The same problem continued with bigger factorials (17!, 18!, 19!, perhaps more?) as well.

Can someone find any sense with this list or is this just natures cruel joke to lead us into desperation?