numbers is a base-2 pseudoprime, and so are all Fermat composites and Mersenne composites. Factorizations The factorizations of the 60 Poulet numbers up to 60787, including 13 Carmichael numbers (in bold), are in the following table. A Poulet number all of whose divisors d divide 2d − 2 is called a super-Poulet number. There are infinitely many Poulet numbers which are not super-Poulet Numbers. Smallest Fermat pseudoprimes The smallest pseudoprime for each base a ≤ 200 is given in the following table; the colors mark the number of prime factors. Unlike in the definition at the start of the article, pseudoprimes below a are excluded in the table. (For that to allow pseudoprimes below a, see ) List of Fermat pseudoprimes in fixed base n For more information (base 31 to 100), see to , and for all bases up to 150, see table of Fermat pseudoprimes (text in German), this page does not define n is a pseudoprime to a base congruent to 1 or -1 (mod n) Which bases b make n a Fermat pseudoprime? If composite is even, then is a Fermat pseudoprime to the trivial base . If composite is odd, then is a Fermat pseudoprime to the trivial bases . For any composite , the number of distinct bases modulo , for which is a Fermat pseudoprime base , is where are the distinct prime factors of . This includes the trivial bases. For example, for , this product is . For , the smallest such nontrivial base is . Every odd composite is a Fermat pseudoprime to at least two nontrivial bases modulo unless is a power of 3. For composite n < 200, the following is a table of all bases b < n which n is a Fermat pseudoprime. If a composite number n is not in the table (or n is in the sequence ), then n is a pseudoprime only to the trivial base 1 modulo n. For more information (n = 201 to 5000), see, this page does not define n is a pseudoprime to a base congruent to 1 or -1 (mod n). When p is a prime, p2 is a Fermat pseudoprime to base b if and only if p is a Wieferich prime to base b. For example, 10932 = 1194649 is a Fermat pseudoprime to base 2, and 112 = 121 is a Fermat pseudoprime