In *Report on the Theory of Numbers*, H.J.S. Smith writes:

"The impossibility of solving [Fermat's] equation has been demonstrated by M. Kummer, first, for all values of $\lambda$ not included among the exceptional primes; and secondly, for all exceptional primes which satisfy the three following conditions:

- That the first factor of H, though divisible by $\lambda$, is not divisible by $\lambda^2$.
- That a complex modulus can be assigned, for which a certain definite complex unit is not congruous to a perfect $\lambda$-th power.
- That $B_{\kappa \lambda}$ is not divisible by $\lambda^3$, $B_{\kappa}$ representing that Bernoullian number $[\kappa \leq \mu-1]$ which is divisible by $\lambda$.

Three numbers below 100, viz. 37, 59, 67, are, as we have seen, exceptional primes. But it has been ascertained by M. Kummer that the three conditions just given are satisfied in the case of each of these three numbers; so that the impossibility of Fermat's equation has been demonstrated for all values of the exponent up to 100. **Indeed, it would probably be difficult to find an exceptional prime not satisfying the three conditions, and consequently excluded from M. Kummer's demonstration.**"

Can anyone cite an exceptional prime that does not satisfy the three conditions?