Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$.

Question

Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$.

I need to use the sequence $a_k=2^{5^{k-1}}$ but not sure how to? Any hints?

Answer 1

The basic idea is finding elements of order 4 modulo successively higher powers of 5. To find an element of order 4 modulo a prime power $p^k$ (when one exists), you can find a primitive root modulo $p^k$ and then raise to the $\frac{\phi(p^k)}{4}$ power. In this case, you should check:

1. 2 is a primitive root modulo every power $5^k$. (This follows from the general result that if $r$ is a primitive root modulo an odd prime $p$, then either $r$ or $r+p$ is a primitive root modulo every power of $p$.)
2. $\phi(5^k) = 4 \cdot 5^{k-1}$.

From this, it follows that $2^{5^{k-1}}$ is an element of order $4$ modulo $5^k$. So choose a large enough value for $k$, and you'll have an element whose residues modulo six consecutive powers of $5$ each have order $4$. Hensel's lemma shows there are two elements of order $4$ in $\mathbb{Z}_5^{\times}$, and you will have found the residues of one of them modulo the first few powers of $5$.