Regarding the equation
13! + 288^2 = (288*274)^2
a further rumination. The Arizal writes that the seven kings that died actually contained 320 sparks. He explains in Ets HaChaim, Heichal Zeir Anpin, Sh'ar רפ"ח Nitzotzin, Perek 5, how the 320 became 325 and from there he gets to 288, but don't ask me how.
However, if you would ask me anyway, here is a derivation. From the above equation, derive
13!*(320/288)^2 + 320^2 = (320*274)^2
Now, observe that (320/288)^2 is very close to 16/13. Both are equal to 1.23, in the decimal approximation.
The principle of the previous post apparently applies to the other equation as well, in opposite fashion.
An alternate Universe, based on the number 16, with 320 sparks of holiness, and 16.986 billion years of age, gave way to our Universe, based on the number 13, with 288 sparks of holiness, and 13.801 billion years of age. The number 13, then, is known as the number of Midot, attributes of G-d, the Midot of Rachamim. A daring thought for the seventh day of Channukah.
