I believe you mean "the number of buckets" not the number of valid hash keys ... there is a catch though ... the m usually depends on n! How exactly, depends, but I beleive in the Perl builtin implementation its 2*n rounded to the nearest power of 2. Which means your o(n/2m) becomes o(n/4n) wich becomes o(1/4) and ... well ... constants are irrelevant in the o() notation thus it's o(1). QED.
The average case of hash lookup even with the naive implementation is o(1), the worst case is O(n) for the naive implementation, but there are ways around that. They make the implementation more complex and insertion more costly, but they are possible. You can rehash once the list in a bucket gets too long or you can use binary trees instead of lists in the buckets.
Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
Read Where should I post X? if you're not absolutely sure you're posting in the right place.
Please read these before you post! —
Posts may use any of the Perl Monks Approved HTML tags:
 a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)

For: 

Use: 
 &   & 
 <   < 
 >   > 
 [   [ 
 ]   ] 
Link using PerlMonks shortcuts! What shortcuts can I use for linking?
See Writeup Formatting Tips and other pages linked from there for more info.

