Determine the set of integers for which n^2 + 19n + 92 is a square.

The solution to this puzzle is as follows.

By the problem,

n^2 + 19n + 92 = p^2, where p is an integer whether positive or negative.

Or, (2n+19)^2 =(2*p)^2 -7

Or, (2p+2n+19)(2p-2n-19) =7*1=1*7 = -7*-1=-1*-7; giving:

(2p+2n+19, 2p-2n-19)=(7,1),(1,7),(-7,-1),(-1,-7)

Solving, we obtain: (p, n) = (2, -8);(2,-11);(-2, -11);(-2,-8); so that the only integers satisfying the equation under reference are given by n = -8 and n = -11.

(It is never mentioned in the problem that n must be positive)

*Edited on ***May 7, 2006, 9:46 am**