User blog comment:Ikosarakt1/Apocalyptic function/@comment-25418284-20130322220331/@comment-5150073-20130323101814

I found that cycle of the last n digits has cycle length 4*5^(n-1)+n-1. For n=1, it gives cycle length 4: ...2, ...4, ...8, ...6. For n=2, the cycle is larger:

2^1 = ...02 2^2 = ...04 2^3 = ...08 2^4 = ...16 2^5 = ...32 2^6 = ...64 2^7 = ...28 2^8 = ...56 2^9 = ...12 2^10 = ...24 2^11 = ...48 2^12 = ...96 2^13 = ...92 2^14 = ...84 2^15 = ...68 2^16 = ...36 2^17 = ...72 2^18 = ...44 2^19 = ...88 2^20 = ...76 2^21 = ...52 2^22 = ...04 2^23 = ...08

Notice that the cycle returns on itself only at 2^22, not 2^21. This pattern observed on the larger cycles, for example, 2^102 has the last digits ...504 instead of expected ...004. However, 2^103 has the last digits ...008, so it returns on the cycle later than it expected, but at right away on its 4*5^(n-1)+n-th member.

As for computing larger cycles, that problem is for future quantum computers that will perform millionplexes (10^1000000) operations per second.