0! + 0^2 = 1^2
1! + 0^2 = 1^2
Altogether, therefore, there are 14 such identities, for 14 numbers n for which (ceiling(sqrt(n!)))^2 - n! is square. They correspond to the 14 Tikkunim of Zeir Anpin and Malchut. The small squares represent the Tikkunim that raise the imperfect factorials to perfect squares.
More specifically, 0! corresponds to the first Samech. The next three identities, for 1!, 4! and 5!, belong to the triple Sephirot not associated to Machzor HaGadol - Yesod, Gevura, and Chesed. From 6! onwards, the identities follow the Tikkunim of Machzor HaGadol. Therefore, the identity for 6! belongs to ה, and 7! + 1^2 = 71^2 is for ו. The 71 at the right-hand side hints at Sanhedrin, in line with the meaning of the Tikkun at ו.
Towards the end, the awesome series of identities for 13! through 16! corresponds to the three Tikkunim of the Sofit letters ץ, ם, Aleph Sofit, and the final Samech.
The identity for 11! corresponds to ת, the start of the count of Arich Anpin. It can be written as (12-1)! + (19-1)^2 = (89*71-1)^2. This hints at the following. From 7! + 1^2 = 71^2 we can learn that size 71 of Beit Din HaGadol derives from the left, from the number 7, the length of the cycle of Din. Conversely, 89 is related to the right, to the cycles of 12 and 19, of Chesed and Yesod.
If one will say that also 71 appears in the identity for (12-1)!, the answer is that the left is included in the right. We will reach a new spiritual level. The emphasis of Sanhedrin will change from Law to Truth. The latter includes, must include, the former. Who thinks we can do without Law, is up to destroy the world. Who would propose that Law can contain the Truth, would also destroy the world.
אורייתא איהי חילא דימינא כמה דאת אמר (דברים לג) מימינו אש דת למו ושמאלא אתכליל בימינא. מאן דעביד ימינא שמאלא ושמאלא ימינא הא איהו כאילו חריב עלמא
At the start of the count of Arich Anpin, the size of Sanhedrin is appropriately increased with one quarter, to 89, in order to encompass five Partzufim rather than four.
עג
