Euler stated many theorems, so it's hard to tell which one Q.1 considers. But none implies: if $b$ is coprime to $w$ and $b^{w-1}\bmod w=1$ then $w$ is prime, and that does not hold. A counterexample is $w=341$ (or other in A001567), $b=2$. Another is $w=561$ (or other in A002997), and any $b$ coprime to $w$.
I think Q.1 is (incorrectly) using Fermat's little theorem, which can be re-stated as: if $b$ is coprime to $w$ and $w$ is prime then $b^{w-1}\bmod w=1$.
If $z=w−1$ what does that actually mean?
It means that either $w$ is prime, or we happen to have used $b$ such that $w$ is a strong pseudoprime to base $b$.
Using the notation in FIPS 186-5 appendix B.3.1 and B.3.2, the strong pseudoprime test works as follows. We want to test some odd integer $w>2$ for primality. We find the uniquely defined $a$ and odd $m$ with $w-1=m\cdot2^a$. Notice that $a\ge1$. We select some $b$ with $1<b<w-1$. Notice that if $w$ is prime then $b$ is coprime with $w$.
Define $z_0=b^m\bmod w$, and for $j>0$ define $z_j={z_{j-1}}^2\bmod n$. By induction, it holds $z_j=b^{m\cdot2^j}\bmod w$. Therefore, by Fermat's little theorem, if $w$ is prime then $z_a=1$.
Therefore if $w$ is prime then ${z_{a-1}}^2\bmod w=1$, which implies $(z_{a-1}-1)(z_{a-1}+1)\bmod w=0$. Therefore if $w$ is prime then $z_{a-1}$ is one of $1$ or $w-1$.
If $w$ is prime and $a>1$ and $z_{a-1}=1$, then similarly $z_{a-2}$ is one of $1$ or $w-1$.
By infinite descent, if $w$ is prime then either $z_0=1$ or there exists some $j$ with $0\le j<a$ and $z_j=w-1$.
By contraposition, if $z_0\ne 1$ and for all $j$ from $0$ to $a-1$ it holds $z_j\ne w-1$ then $w$ is not prime.
The strong pseudoprime test applies the above theorem. The simplest implementation computes all the $z_j$ for $j$ from $0$ to $a-1$ per the definition given above, and tests if it holds $z_0\ne 1$ and $z_j\ne w-1$ for all these $j$. In the affirmative, it's concluded that $w$ is not prime. Otherwise that strong pseudo prime test to base $b$ is inconclusive: $w$ could be prime, or not. In the later case, $w$ is said to be a strong pseudoprime to base $b$. An example is $w=2047$ (or other in A001262), $b=2$.
In FIPS 186-5 B.3.1 and B.3.2, and commonly, it is used three independent "early abort" optimizations of that simplest algorithm for the strong pseudoprime test, none of which changing that test's outcome. Computation can be aborted as soon as it's found any one of:
- $z_0=1$, with the strong pseudoprime test inconclusive.
- $z_j=w-1$, with the strong pseudoprime test inconclusive.
- $z_j=1$ for $j\ne 0$ and $z_{j-1}=w-1$, with the strong pseudoprime test concluding that $w$ is not prime.
A Miller-Rabin test performs a number of strong pseudoprime tests for random choice of base $b$, stopping when a strong pseudoprime test concludes that $w$ is composite (which also is the conclusion of the Miller-Rabin test). Otherwise the Miller-Rabin test concludes that $w$ is probably prime.
The enhanced Miller-Rabin test of B.3.2 performs the same, excepts that
- for each $b$ chosen it's computed $g=\gcd(w,b)$ and if $g\ne1$ the enhanced Miller-Rabin test terminates concluding that $w$ is a composite with non-trivial factor $g$.
- when the strong pseudoprime test with (at least) the last of the above three optimizations finds a $z_j$ allowing to conclude that $w$ is not prime, that is when for $j>0$ it holds $z_j\ne w-1$ and $z_{j+1}=1$ (as detected by said third optimization), or when it's reached $j=a-1$ with $z_j\ne w-1$, then it's computed $g=\gcd(w,z_j)$, and:
- if $g\ne1$, the enhanced Miller-Rabin test terminates concluding that $w$ is a composite with non-trivial factor $g$.
- if $g=1$, the enhanced Miller-Rabin test terminates concluding that $w$ is a composite not a prime power. I refer to @poncho's proof of that later property.
