The author delves into the intriguing sequence of the last two digits of 2^n, revealing the finite nature of the sequence where n == 3, 6, 10, 11, or 19 (mod 20), and the oddity of tens and hundreds digits. Surprisingly, after extensive examination up to 50000, no new terms have been discovered. An in-depth exploration of digits uncovers that it was only once necessary to search even to the 18th digit for 2^12106, a 3645-digit number. A program is provided in Mathematica by Harvey P. Dale to handle this mathematical curiosity. Charles R Greathouse IV’s PARI code further explores the oddity aspect of the sequence.
https://oeis.org/A068994