Integers of the form (1+2x+4x^3)/(x+n)

[Giovanni Resta]

zak seidov wrote:

> Not submitted yet
> My Qs to gurus:
> 
> No solutions for n>10?

I do not understand that. It seems to me that there
are solutions for any n.
Maybe there is something that is not clear to me.

For example, for n>10, you got integers for
n=11 x={-5356, -1080, -16, -12, -10, -6, 1058, 5334}
n=12 x={-6947, -1399, -377, -107, -85, -31, -17, -13, -11, -7, 7, 61,
    83, 353, 1375, 6923}
n=13 x={-8826, -1272, -20, -14, -12, -6, 1246, 8800}
and so on (I only checked range -10^5...10^5).

In particular you always obtain an integer at least
for x=1-n, and x=-1-n
because the denominator x+n is then equal to +/- 1.

Apart that, how do you proved that the only solutions
for, say n=10, are {-9,-11,4009,-4029} ?

giovanni