very difficult congruence

[logistic_guy]

here is the question

Find all solutions of \(\displaystyle x^4 - 13x^3 + 11x - 3 \equiv 0 \ (mod \ 7^8)\).

my attemb
is there a way to solve this or i've to use a computer?

 

[fresh_42]

What remains after you eliminated the solution [imath] x=-1\equiv 5,764,800 \pmod{7^8}[/imath]?

 

[logistic_guy]

What remains after you eliminated the solution [imath] x=-1\equiv 5,764,800 \pmod{7^8}[/imath]?

thank

i don't understand what you mean by eliminating the solution, but i see you get lucky to find an negative integer solution make the polynomial equal zero. what happen if the polynomial don't have negative integer solutions like this

\(\displaystyle x^9 + 13x^3 - x + 100336 \equiv 0 \ (mod \ 17^9)\).

 

[fresh_42]

There are no golden rules that I know of. [imath]f(x) \equiv 0 \pmod{p^n} [/imath] only means that [imath] p\,|\,p^n\,|\,f(x) [/imath] from where you can start. Things quickly become complicated if you cannot factor [imath] f(x) .[/imath]