What are the factors that make an application competitive for postdoc positions in pure mathematics at top tier universities (e.g. MIT, Harvard, Princeton, UC Berkeley, Stanford) in the US?
Clearly, one needs
In what proportion are the above needed? E.g. how many good articles, how many courses taught, etc. make an application competitive?
Is there any other important factor I haven't mentioned above?
There's no simple answer. In particular, there are no quantitative standards for how many papers you need. For postdoctoral research positions in mathematics at top departments in the U.S., it works roughly like this. (For those from other fields, note that these positions are filled by departmental search committees, not individual professors, and they often involve some light teaching but are focused on research. The postdocs are independent researchers, not under the direction of any professor.)
Teaching experience and the teaching statement don't matter much for these jobs. The department will want to know that you aren't a truly terrible teacher, so you won't raise tons of complaints or create a serious problem if you teach a class, but they won't really care whether you are an excellent teacher or just minimally adequate. (At best that would serve as a tie breaker between otherwise equally impressive candidates.) Just write something sensible and uncontroversial for your teaching statement and don't worry about it much.
Quality matters much more than quantity for research papers, and trying to write more papers at the cost of writing worse papers is generally a bad trade-off. In practice, the typical number of papers varies a little by subfield. In some cases (such as theoretical CS), it would be very unusual to have only one paper, and it had better be amazing if you want to get a job that way; in some other subfields, having half a dozen papers might make people suspicious regarding their quality. To gauge how your application compares, you can try looking at the CVs of people in your area who have recently started the sort of job you'd like, but keep in mind that counting papers won't reveal their quality.
Strong recommendation letters are absolutely crucial, and getting good letters is worth a lot of time and thought as well as preparation in advance, for example by talking with potential letter writers so they are familiar with you and your work long before you apply for jobs. Choose your recommenders carefully, and don't limit yourself only to people at your university.
