Why is $561$ missing from this list?

Question

On the MathWorld page:

http://mathworld.wolfram.com/FermatPseudoprime.html

in the first table, I expect to see $561$ on every line, but it is not on the line for base $3$.

When you click on the link to the OEIS page, it also is missing from the list. Since $561$ is a Carmichael number, I expected it to be there. Is this a typo (and if so, how do I report it)? If not, what am I missing? Certainly $3^{561} \equiv 3 \pmod{561}$; is there a different definition of "Fermat pseudoprime" that leaves $561$ out?

Answer 1

$3$ divides into $561$, so it is not true that $3^{560} \equiv 1 \pmod {561}$. The definition on the page calls for this, not for $3^{561} \equiv 3 \pmod {561}$

Answer 2

The definition is that a base $a$ pseudoprime $n$ satisfies $a^{n-1} \equiv 1 \mod a$. But $3^{560} \equiv 375 \mod 561$.

Answer 3

That post says that $q$ is a Fermat pseudoprime to base $b$ if any only if:

$$b^{q-1}\equiv 1\pmod q$$

It is possible for $b^{q}\equiv b\pmod{q}$ without $b^{q-1}\equiv 1\pmod{q}$, if $b,q$ are not relatively prime.

Answer 4

Carmichael numbers are composite numbers that are not pseudoprime for all factors of that number. So $561=3\times 11 \times17$ is not a pseudoprime base 3, 11 and 17. Carmichael numbers only needs to be a psuedoprime for a base that is coprime to the Carmichael number.