Fermat pseudoprime

An odd composite number $N$ is a Fermat pseudoprime to a coprime base $b$ when

A Fermat pseudoprime is relative to its base. For example, $341=11\cdot 31$ passes base $2$, but base $3$ yields $3^{340}\equiv 56(\mathrm{mod}341)$, a Fermat witness. A Carmichael number has the stronger all-coprime-base property. This explains why finding one deceptive base and proving universal deception require different arguments.

Write the even number $N-1$ as $2^{s}m$ with $m$ odd. An odd composite number $N$ is a strong pseudoprime to a coprime base $b$ whenor $b^{2^{j}m}\equiv -1(\mathrm{mod}N)$ for some $0\le j<s$.

Let $N=pq$ for distinct odd prime numbers $p,q$ with $\gcd (p-1,q-1)=10$. Exactly $100$ unit bases make $N$ a Fermat pseudoprime, and exactly $50$ of these make it a strong pseudoprime. The Fermat bases form $C_{10}\times C_{10}$ under the Chinese remainder theorem for unit groups. If $N-1=2^{s}m$, then $5\mid m$ and $m$ is odd, so the $m$th-power map sends five elements of each $C_{10}$ to $1$ and five to $-1$. The strong condition requires equal signs in both components, giving $5^2+5^2=50$.