  We consider the distribution of the largest prime divisor of the integers in
the interval $[2,x]$, and investigate in particular the mode of this
distribution, the prime number(s) which show up most often in this list. In
addition to giving an asymptotic formula for this mode as $x$ tends to
infinity, we look at the set of those prime numbers which, for some value of
$x$, occur most frequently as the largest prime divisor of the integers in the
interval $[2,x]$. We find that many prime numbers never have this property. We
compare the set of "popular primes," those primes which are at some point the
mode, to other interesting subsets of the prime numbers. Finally, we apply the
techniques developed to a similar problem which arises in the analysis of
factoring algorithms.
