Vandiver's Criteria

Let $p$ be an Irregular Prime, and let $P=rp+1$ be a Prime with $P<p^2-p$. Also let $t$ be an Integer such that $t^3\not\equiv 1$ (mod $P$). For an Irregular Pair $(p, 2k)$, form the product

$$Q_{2k}=t^{-rd/2}\prod_{b=1}^m (t^{rb}-1)^{b^{p-1-2k}},$$

where

$$m = {\textstyle{1\over 2}}(p1-1)$$

$$d = \sum_{n=1}^m n^{p-2k}.$$

If ${Q_{2k}}^r\not\equiv 1$ (mod $P$) for all such Irregular Pairs, then Fermat's Last Theorem holds for exponent $p$.

See also Fermat's Last Theorem, Irregular Pair, Irregular Prime

References

Johnson, W. ``Irregular Primes and Cyclotomic Invariants.'' Math. Comput. 29, 113-120, 1975.