Can a student write a paper and send it to a professor review?

3$\begingroup$ Certainly, but it would be a good idea to talk to your professors about it for advice in any case. You might also read the webpage research.microsoft.com/enus/um/people/cohn/Thoughts/… $\endgroup$ – Bjorn Poonen May 20 '10 at 21:04

$\begingroup$ That link is to a webpage by Henry Cohn, by the way. $\endgroup$ – Bjorn Poonen May 20 '10 at 21:05

18$\begingroup$ Make sure it is not a proof of Fermat's last theorem or the Goldbach conjecture. $\endgroup$ – KConrad May 20 '10 at 21:16

2$\begingroup$ In that case,we need to put the kid under Secret Service protection on his way to the Pentagon.KConrad. $\endgroup$ – The Mathemagician May 20 '10 at 21:33

10$\begingroup$ I have edited the title to remove the gendered pronoun. $\endgroup$ – Theo JohnsonFreyd May 20 '10 at 23:40
Sure, why not?
But let me suggest a better plan. What you need is a faculty mentor. If you have a favored topic that you are working on, then first simply go and speak about it with a knowledgable professor about it in person. It will be extremely helpful to you to have a faculty mentor in your mathematical work. You can explain your idea and result and discuss it with him or her. Please try to listen to what the professor says about your idea. In the best situation, he may be able to guide you by providing some ideas that will help you to extend or improve your results. (In other circumstances, you may learn that the result is already known, or trivial or wrong in some way, which would be extremely important for you to know if it is true.)
In the best circumstance, the professor may encourage you to continue on the project and write a paper. In this case, most professors would also be willing to review the final draft, or even to supervise an undergraduate research experience based on the topic. Usually writing a mathematics paper involves many more drafts and revisions than novices expect, particularly when the author does not have much experience with mathematical writing. The help provided by a faculty mentor will enable you to bring your work to a higher level.
Yes, this has been done and in fact there is an award for undergraduate mathematics research the Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student

7$\begingroup$ Generally they'd have a supervisor of some sort though correct? So you don't, as one graduate student once put it, "Go off into lala land" i.e. lose all focus. $\endgroup$ – Michael Hoffman May 20 '10 at 21:16

1$\begingroup$ I have looked at a few of the winners and they had a supervisor at some point. $\endgroup$ – Kristal Cantwell May 20 '10 at 21:58
Permit me to draw attention to a relatively new journal, Involve—A Journal of Mathematics. Snippets from About the Journal:
"Involve is dedicated to showcasing and encouraging high quality mathematical research involving students (at all levels). ... Submissions in all mathematical areas are encouraged. However, each manuscript should include a minimum of 1/3 student authorship. ... Involve is a publication in between the extremes of purely undergraduate research journals, which in general are aimed at undergraduate audiences, and mainstream research journals."
I would guess that nearly all of the papers that have appeared in its first nine issues reflect work supervised by a faculty mentor, and a good percentage are coauthored with that faculty member (and not infrequently with more than one student).