TITLE: Solve the diophanic equation $y^2 = x^4 -4x^3 + 6x^2 -2x +5$, where $x,y \in \mathbb{Z}$
QUESTION [5 upvotes]: Solve the diophanic equation $y^2 = x^4 -4x^3 + 6x^2 -2x +5$.
Methods I know:
1) look modulo p for some prime p, when using this method I almost always conclude there are no solutions, so i don't think it is handy to use this in this particular case.
2) Combinations of factorization and estimation: factorize your specific function and Search for an upper bound.
I used 2), and i get in a little bit of trouble.
I tried factorize the right hand side, but did not come any further then:
$y^2 - 10 = (x-5)(x^3 + x^2 - x - 3)$. 
I don't seem to get an upper bound here..... :(.
Any hints on going to the right direction or on using a new method trying to solve this probelem?
Kees

REPLY [6 votes]: We may suppose that $y\geq 0$. Now
$$x^4-4x^3+6x^2-2x+5=(x-1)^4+2x+4$$
Hence we have
$$(y-(x-1)^2)(y+(x-1)^2)=2(x+2)$$
This imply that $(x-1)^2\leq (x-1)^2+y\leq 2|x|+4$
and it is easy to finish.