There are exactly four positive integers such that (n + 1)^2/(n + 27) is an integer. Compute the largest such n.

There are 4 integer value of n as follows:

n ==13, 121, 288, 625 [The largest such number]

The above 4 numbers are NOT correct! The following 4 numbers are correct:

n = 25, 142, 311, 649

Therefore, the largest n=649