Every composer knows the anguish and despair occasioned by forgetting ideas which one had no time to write down. (Hector Berlioz)

There were many occasions early in my career when I read, heard about, or stumbled upon some neat mathematical trick or argument, and thought I understood it well enough that I didn’t need to write it down; and then, say six months later, when I actually needed to recall that trick, I couldn’t reconstruct it at all. Eventually I resolved to write down (preferably on a computer) a sketch of any interesting argument I came across – not necessarily at a publication level of quality, but detailed enough that I could then safely forget about the details, and readily recover the argument from the sketch whenever the need arises.

I recommend that you do this also, as it serves several useful purposes:

  1. It makes the argument permanently available to you in the future, and may eventually be helpful in your later research papers, lecture notes, teaching, or research proposals.
  2. It gives you practice in mathematical writing, both at the technical level (e.g. in learning how to use TeX) and at an expository or pedagogical level.
  3. It tests whether you have really understood the argument on more than just a superficial level.
  4. It frees up mental space; you no longer have to remember the exact details of the argument, and so can devote your memory to learning newer topics.

Once you have written up such a sketch, you might consider making it available (e.g. on your web site), even if it does not rise to the level of originality and depth required for a publishable paper.

For somewhat similar reasons, if you have an incomplete (or otherwise unsatisfactory) argument for a problem that you are working on, and you are planning to abandon it, you may still wish to write an informal sketch of it just for yourself (giving barely enough details to allow you to readily reconstruct the whole thing later on), and store it somewhere on your computer, just in case you find you have need for it some time in the future. (Of course, I also recommend some way of backing up your computer files, e.g. by using a cloud-based file storage system such as Dropbox.)