Smallest Strong Fibonacci Pseudoprime of Type I

Theorem
The smallest strong Fibonacci pseudoprime of type I is $443 \, 372 \, 888 \, 629 \, 441$.

Proof
Let $N := 443 \, 372 \, 888 \, 629 \, 441$, to save writing it out in full each time.

From the definition of a strong Fibonacci pseudoprime of type I:

We have that:
 * $N = 17 \times 31 \times 41 \times 43 \times 89 \times 97 \times 167 \times 331$

Of these, we see that:

Thus there are $4$ (an even number) of prime factors of $N$ of the form $4 n - 1$.

We note that:

Thus for all $p \divides N$, we have that $\paren {p^2 - 1} \divides \paren {N - 1}$.

From Difference of Two Squares we have that:
 * $p^2 - 1 = \paren {p + 1} \paren {p - 1}$

and so for all $p \divides N$:
 * $\paren {p - 1} \divides \paren {N - 1}$

We also have that $N$ is square-free.

Thus by Korselt's Theorem, $N$ is a Carmichael number.

Now we also have that:

Thus we have that:
 * $\forall p \divides N: 2 \paren {p + 1} \divides \paren {N - 1}$

This is actually stronger than the conditions for which $N$ is a strong Fibonacci pseudoprime of type I:

It can be established by an exhaustive search that there are no smaller Carmichael numbers with this property.

Hence the result.