For how many positive integers n is n^2 + 96 is a perfect square?

Math Central

Quandaries & Queries

Question from Rajesh, a student:

For how many positive integers n is n² + 96 is a perfect square?

Hi Rajesh,

Suppose that n² + 96 = m² for some positive integer m, then

m² - n² = 96
(m - n)(m + n) = 96 = 2⁵ × 3

96 has 12 divisors which come in pairs and since m - n < m + n they are

m - n m + n

1 96

2 48

3 32

4 24

6 16

8 12

For each of the 6 cases solve for m and n and verify which pairs, if any, satisfy your requirements.

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.