I have to find all the solutions to the congruence $x^2 = -6(\text{mod}\, 625)$ using Hensel's lemma and I find it quite difficult. If anyone could point me to the solution I'll be grateful, thanks in advance.
[Math] Find all congruence solutions using Hensel’s Lemma
congruenceselementary-number-theorynumber theory
Best Answer
My first post here, so please excuse me if I'm not doing this right.
In response to Stefan4024's post above, $5^4 = 625$. You erroneously wrote $5^3 = 625.$
To solve the problem, you only need to lift twice.
After finding $x = \pm 2 ~(\text{mod } 5)$ or $x = 2, 3$ to be the solutions of $f(x) = 0 ~(\text{mod }5)$, the solutions can be lifted $(\text{mod }5^2)$ or $(\text{mod }25).$
After that, just a single lift to (mod $5^4$) or (mod $625$) is required. This is because one can lift a solution (mod $p^k$) to (mod $p^{m+k}$) as long as m is less than or equal to $k$. In the first lift (from $5$ to $25$), $m = k = 1$, and in the second lift (from $25$ to $625$), $m = k = 2$.
Doing this for both solutions $x = 2$ and $x = 3 ~(\text{mod }5)$ gives the respective solution sets for $f(x) = 0 ~(\text{mod }625)$ as :
$$x = 162 + 625n$$
and
$$x = 463 + 625n$$
where $n$ can take any integer value.
The solutions can be expressed more compactly as:
$$x = 162 \text{ or } 463 ~~(\text{mod }625)$$