We show that if there are no three consecutive powerful numbers then there is an infinite sequence of primes \( p\) for which \( p^2\) does not divide \( 2^p-2\), and so the first case of Fermat's Last Theorem is true for exponent \(p\).
We show that if the first case of Fermat's Last Theorem is false for prime \( p\) then \( p\) divides the numerator of \( B_{p-1-n}\) for all integers \( n\leq (\log p/\log\log p)^{1/2}\).