Note that the memorability of the last few digits is a direct consequence of the way you constructed the number (yes, it's 12345^5). The last 4 digits of n^5 when n ends in 5 are always among {0625,1875,3125,4375,6875,8125,9375}. If the digit before the 5 is even you always get one of {0625,3125,5625,8125}. At the very least, the final 3 digits are always a multiple of 125 and you probably recognize all of those.

Still, even a completely random number typically has lots of little patterns in it to help this kind of memorizing. For instance, I just generated a random 20-digit number: 66474746605022249923. The first things that occur to me, looking through it in order:

• 66 4747 -- two pairs of repetitions
• 466 -- overlapping 46 (one less than the 47 we just had) and 66 (same as first two digits, and a pair)
• 050 -- symmetrical, all multiples of 5 (of which there aren't many among the digits :-))
• 222 --- threefold repetition
• 499 -- one less than 500
• 23 -- not a particularly interesting number but e.g. the number of chromosome pairs you have.

(I also noticed in passing that 60502 is reminiscent of 6502, the processor in the first few computers I used. Lovely instruction set. Having some overlap between the features one notices is useful because it makes it easier to remember what order things come in.)

I tried the obvious experiment: after writing the above, could I look away from it and reproduce my 20-digit number? Why yes, I could; and still could a couple of minutes later. I think I'd find things like reversing the digits quite painful, though.