I am reading Carmichael numbers in Algorithms book by Sanjoy das gupta at followinglink page 38

<http://www.cse.iitd.ernet.in/~naveen/courses/CSL630/all.pdf>

Carmichael numbers The smallest Carmichael number is 561. It is not a prime: 561 = 3\*11\* 17; yet it fools the Fermat test, There is a way around Carmichael numbers, using a slightly more refined primality test due to Rabin and Miller. Write N - 1 in the form 2^t \* u. As before we'll choose a random base a and check the value of a ^N-1 mod N. Perform this computation by first determining a^u mod N and then repeatedly squaring, to get the sequence:

```
a^u mod N; a^2u mod N; : : : ; a^2tu = a^N-1 mod N:

```

If a^N-1 not equal to 1 mod N, then N is composite by Fermat's little theorem, and we're done.

But if a^N-1 is equal to 1 mod N, we conduct a little follow-up test: somewhere in the preceding sequence, we ran into a 1 for the first time. If this happened after the first position (that is, if a^u mod N is not equal to 1), and if the preceding value in the list is not -1 mod N, then we declare N composite.

In the latter case, we have found a nontrivial square root of 1 modulo N: a number that is not 1 mod N but that when squared is equal to 1 mod N. Such a number can only exist if N is composite. It turns out that if we combine this square-root check with our earlier Fermat test, then at least three-fourths of the possible values of a between 1 and N - 1 will reveal a composite N, even if it is a Carmichael number.

Following questions on above text.

Question 1:

I am not able to follow below text 

> 
> But if a^N-1 is equal to 1 mod N, we conduct a little follow-up test:
>  somewhere in the preceding sequence, we ran into a 1 for the first
>  time. If this happened after the first position (that is, if a^u mod N
>  is not equal to 1), and if the preceding value in the list is not -1
>  mod N, then we declare N composite.
> 
> 
> 

What does above statement mean? Any example will help.

Question 2:

What does author talking about square root here in last paragraph first two lines?

Thanks