Question

Prove that the square of an interger is not of the form 3n+2 where n is an integer

Answer 1

If the square of the integer n is (3n+2), then
n² = 3n + 2

That is,
n² - 3n - 2 = 0

Solve with the quadratic formula.
$n= \frac{1}{2} [3 \pm \sqrt{9+8} ] \\ n= \frac{1}{2}(3+ \sqrt{17} ) \,\, or \,\, n= \frac{1}{2}(3- \sqrt{17} )$

The solutions for n are rational numbers, not integers.

Answer:
The square of an integer should be an integer, therefore the square of n is not of the form (3n+2).