Question

The number of natural numbers n for which n²+96 is perfect square is.........​

Answer 1

492 - 482 = 2401 - 2304 = 97

Therefore any possible answer must be less than 49.

The positive integers n for which n2 + 96 = y2 where y is an integer are

2, 5, 10, and 23

2 and 10

5 and 11

10 and 14

23 and 25

a somewhat more formal apprPacheco is to say that (n+1)^2 - n^2 = 96

(n+2)^2 - n^2 = 96

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.

.

(n+9)^2 - n^2 = 96

there is no need to go beyond n+9 because the only possible solution is n<1

so we can solve for n in those cases were 2i + i^2 is even

This occurs for n= 2, 5, 10, and 23

it looks like this

4n + 4 = 96

8n + 16 = 96

12n + 36 = 96

16n + 64 = 96

enjoy !