The following is an explanation of our extension of Brocard's Problem or Brocard's Enigma. Regarding the following 14 equations
0! + 02 = 12
1! + 02 = 12
4! + 12 = 52
5! + 12 = 112
6! + 32 = 272 = (3*9)2
7! + 12 = 712
8! + (3*3)2 = 2012 = (3*67)2
9! + (9*3)2 = 6032 = (9*67)2
10! + 152 = 19052 = (15*127)2
11! + 182 = 63182 = (18*351)2
13! + 2882 = 789122 = (288*274)2
14! + 4202 = 2952602 = (420*703)2
15! + (16*29)2 = 11435362 = (16*71471)2
16! + (64*29)2 = 45741442 = (64*71471)2
I postulated almost seven years ago that
D(n) = (ceiling(sqrt(n!)))2 - n!
cannot be a square number if n is greater than 16. What does that mean?
For example, take n=6. sqrt(6!) = sqrt(720) = 26.83282. The ceiling of a number is the smallest integer number above it, or equal to it. In our case, ceiling(sqrt(6!)) = 27. The square of 27 is 729, which is 9 more that 6!. So, D(6) = 9, and it is a square, of 3. Brocard's observation, in our language, is that D(n) = 1 for n = 4, for n = 5, and for n = 7, and after that, seemingly, D(n) ≠ 1 forever.
It happens that D(n) is a square for 14 out of the first 17 integer numbers, detailed above. And after that it is a square no more, according to the postulate, not even once. If the above are the only 14 cases of n for which D(n) is a square, D(n) is equal to 1 exactly three times. Thus, if the postulate is true, it solves the problem of Henri Brocard, and proves the conjecture of Paul Erdős.
Regarding proofs, we know from Erdős, among others, that D(n) ≠ 0, for n > 1. As far as I know, for the remainder, the problem is open. For some more info, see this.
