I have two equations which are: $p^3+k\equiv0 (mod \quad h) $ and $(3p^2+3mp+m^2)m\equiv 0(mod \quad h)$ where $k,h,m >0$ and $p\ge0$ and $h\nmid m$
I need to show for given k,m,h and for all $p\ge 0$ there is no solution for both equation at same time. i.e. when I try by programatically for k=5 and m=1 there isn't any solution for all $p\ge0$
$p^3+5\equiv0 (mod \quad 2) $ and $(3p^2+3p+1)\equiv 0(mod \quad 2)$
is there any way to show this?
edit : when h is prime I guess we can check by euler criterion checking $((3mp)^2-12m^2)^{\frac{h-1}2}$ if it is equvilant to -1 for mod h, there is no solution for this equation. if it is true what do we do when h is not prime?