In my article “Patterns in the Last Digits of the Positive Powers of Five” I noted that the positive powers of five modulo 10^{m} cycle with period 2^{m-2}, m ≥ 2, starting at 5^{m}. In this article, I’ll present my proof, which has two parts:

- Part 1 shows that the powers of five
**mod 2**cycle with period 2^{m}^{m-2}, m ≥ 2,**starting at 5**.^{0} - Part 2 shows that the powers of five
**mod 10**cycle with the same period as the powers of five mod 2^{m}^{m},**starting at 5**.^{m}

The highlight of my proof is in part 1, where I derive a formula to show that the period, or order, of 5 mod 2^{m} is 2^{m-2}. While it is in general not possible to derive a formula for the order of a number, I’ll show it *is* possible for the powers of five mod 2^{m} — **due to a hidden, binary structure I’ve uncovered**.

Continue reading “Cycle Length of Powers of Five Mod Powers of Ten”