  For a fixed integer a>1, we suggest that the probability of nullity of the
p-Fermat quotient q(p,a) is much lower than 1/p for any arbitrary large prime
number p. For this we use various heuristics, justified by means of numerical
computations and analytical results, which may imply the finiteness of the
q(p,a) equal to 0 and the existence of integers a such that q(p,a) is different
from 0 for all p. However no proofs are obtained concerning these heuristics.
