  A famous conjecture bearing the name of Vandiver states that $p \nmid h_p^+$
in the $p$ - cyclotomic extension of $\Q$. Heuristics arguments of Washington,
which have been briefly exposed in Lang (1978), p. 261 and Washington (1996),
p. 158 suggest that the Vandiver conjecture should be false if certain
conditions of statistical independence are fulfilled. In this note, we assume
that Greenberg's conjecture is true for the \nth{p} cyclotomic extensions and
prove an elementary consequence of the assumption that Vandiver's conjecture
fails for a certain value of $p$: the result indicates that there are deep
correlations between this fact and the defect $\lambda^- > i(p)$, where $i(p)$
is like usual the irregularity index of $p$, i.e. the number of Bernoulli
numbers $B_{2k} \equiv 0 \bmod p, 1 < k < (p-1)/2$. As a consequence, this
result could turn Washington's heuristic arguments, in a certain sense into an
argument in favor of Vandiver's conjecture.
