Concrete mathematics : a foundation for computer science

114 NUMBER THEORY
The binary representation also shows us how to derive another formula,
E~(TI!) = n-Y2(n) , (4.24)
where ~z(n) is the number of l’s in the binary representation of n. This
simplification works because each 1 that contributes 2”’ to the value of n
contributes 2”-’ + 2mP2 + . . .+2’=2”-1 tothevalueofcz(n!).
Generalizing our findings to an arbitrary prime p, we have
(4.25)
by the same reasoning as before.
About how large is c,(n!)? We get an easy (but good) upper bound by
simply removing the floor from the summand and then summing an infinite
geometric progression:
e,(n!) < i+l+n+...
P2 P3
= 11 ,+i+$+...
P (
-n P
- -P P-1 0
n
=p_l.
1
For p = 2 and n = 100 this inequality says that 97 < 100. Thus the upper
bound 100 is not only correct, it’s also close to the true value 97. In
fact, the true value n - VI(~) is N n in general, because ~z(n) 6 [lgnl is
asymptotically much smaller than n.
When p = 2 and 3 our formulas give ez(n!) N n and e3(n!) N n/2, so
it seems reasonable that every once in awhile e3 (n!) should be exactly half
as big as ez(n!). For example, this happens when n = 6 and n = 7, because
6! = 24. 32 .5 = 7!/7. But nobody has yet proved that such coincidences
happen infinitely often.
The bound on e,(n!) in turn gives us a bound on p”~(~!), which is p’s
contribution to n! :
Gin!) < pw(P-‘) .
P
And we can simplify this formula (at the risk of greatly loosening the upper
bound) by noting that p < 2pP’; hence pn/(Pme’) 6 (2p-‘)n/(pp’) = 2”. In
other words, the contribution that any prime makes to n! is less than 2”.