$(1$ point$)$ The Miller$-$Rabin primality test is based around the following observation.

If $p$ is prime and $x^2-=1,$modp then $x-=+-1,$modp.

Note that $x-=-1,$modp and $x-=p-1,$modp mean the same thing.

We will use the Miller$-$Rabin test to test $n=2905$ for primality. To do this we will closely examine

$a^{n-1}$modn for various integers $a.$

First we divide out all of the $2\text{'}$s from $n-1.$

We can write $n-1=2^{u}r$ where

$u=$

is maximal, and

$r=$

Now we randomly select some $ain{Z}_n,$ such as $a=2823.$

Initially we compute

$a^r-=,$modn.

If this number is $1$ then the test is inconclusive and another $a$ is selected $($up until the number of

$a^{\text{'}}s$ determined by the security level$).$

Otherwise we compute the list of values $a^{2r},a^{4r},$dots,$a^{2^{i}r},$dots,$a^{2^{u}r}.$ Note $a^{2^{u}r}=a^{n-1}.$

Please enter this list for $n=2905$ and $a=2823$ in comma separated format.

When the list is computed one of a few things could happen.

The value $1$ never occurs on the list.

The value $-1-=p-1,$modp occurs on the list, immediately followed by $1.$

The value $1$ occurs in the list, but preceded by $-1.$

Which one of these possibilities holds in our case $($enter the number$)?$

What conclusion should we draw from what we have done?

$n$ is prime.

$n$ is composite.

The test is inconclusive.