Here is a 3-paragraph summary for mathopen.com:
**The Last Four Digits of 16¹⁶**
A fun mathematical curiosity recently made the rounds on the math-fun mailing list: the number 16 raised to the 16th power ends in the digits 1616. Similarly, 499 raised to the 499th power ends in 499499. These patterns raise a natural question: are there other numbers where raising a number to its own power produces a result ending in that number repeated twice?
To investigate, mathematicians look at this problem through the lens of modular arithmetic. The question becomes: for which values of n does n^n end in the digits nn (the number n written twice)? This requires analyzing the last digits of large powers, which cycles in predictable patterns. The search can be extended to numbers with more digits as well, turning a simple curiosity into a rich number theory problem.
What makes this problem especially appealing is how accessible it feels while still offering genuine depth. Anyone can verify the original examples with a calculator, but finding all solutions requires careful mathematical reasoning about remainders and repeating patterns. It is exactly the kind of puzzle that invites exploration at multiple levels, making it a perfect example of recreational mathematics at its best.