# JMO question---

Nishant Vora IIT Patna
8 years ago
Hello Student

n6 +3n5_5n4_15n3+4n2+12n+3 =
= n5(n+3) –5n3(n+3)+4n(n+3) + 3
=(n+3)[n5 –5n3+4n] + 3
=n(n+3)[n4-5n2+4] + 3
=n(n+3)(n2-4)(n2-1) + 3
=(n-2)(n-1)n(n+1)(n+2)(n+3) + 3

So this can not be perfect square

Thansk
mycroft holmes
272 Points
8 years ago
We will prove that this expression is of the form 4k+3 and hence not a perfect square.

$4n^2+12n$ is a multiple of 4. Hence we only need to prove that the following expression is a multiple of 4:

$n^6+3n^5-5n^4-15n^3 = n^6-n^5-n^4+n^3 + 4(n^5-n^4-4n^3)$

So, we only have to prove that for all n $n^6-n^5-n^4+n^3$ is divisible by 4.

If n is a multiple of 4, we are done. Else we write the expression as $n^4(n^2-1)-n^3(n^2-1)$

We know that if n is not even, then n2-1 is divisible by 4 and hence in this case too the expression is a multiple of 4.

Thus, the polynomial is of the form 4k+3 for all n, and therefore not a perfect square