This was a test to prove or deny, that during a REM phase, a memory reshuffling is taking place (sorting recent memories into relevant parts of the brain, making connections between them etc). To get a reliable proof, I had to "summon" a piece of information which managed to meet certain criteria:
1) I encountered this information at least once in my life
2) It has to be something which is easy to check and requiring precision at the same time
3) And the most important - you shouldn't be able to remember this in normal situation (which I certainly couldn't - at that time the most I remembered were 5 numbers after a comma).
All happened pretty spontaneously, during one dream I achieved alertness (I was climbing a huge, curved stairway in abandoned building), and somehow decided that my dream is stable enough to pull this off. I knew that in most of cases, focusing too much on something leads to quick awakening, so I ascertained that I have few "seconds" as limit. I imagined the number and the extension I knew of (3,14159), and the rest started to form in my mind - coming to abrupt end after the 10th number. Instantly after I awakened, I rushed to my desk, grabbed a piece of paper (used bus ticket), and wrote the number I remembered a seconds before: 3,1415926535.