Question

(n^4 + n^3 + 1) is a square number for n positive integer. how many of n be satisfied ?

Answer 1

show that below inequality holds for all \(n\gt 2\)
\[(2n^2+n-1)^2~\lt~ 4(n^4+n^3+1)~\lt ~(2n^2+n)^2\]

Answer 2

you can show that trivially. that proves that 4(n^4+n^3+1) is trapped between two "consecutive" perfect squares. so it cannot be a perfect square.

Answer 3

thanks a ton @rational
that make sense to me

Answer 4

np:)

Answer 5

btw we need to check n=1,2 manually

Answer 6

yes, only n = 2 is win :D

Answer 7

yepp

Answer 8

next question is in new posting :D