The last four digits of 16¹⁶

On the math-fun mailing list, Dick Hess posted “a couple of curiosities”: A speaker at G4G16 noted that \(16^{16}\) ends in \(1616\).A friend sent me this: \(499^{499}\) ends in \(499499\). Are there any other cases including numbers with more digits? That does make me curious! I’ve kept that email in my inbox since March, hoping to have a go at finding more examples when I get some free time. Nobody else responded to the email, which is usually a sign that an answer is hard to find. At first glance, it might seem really hard to even check these facts: \(16^{16}\) is a very big number, and \(499^{499}\) must have hundreds of digits. Can computers even work them out accurately? This reminds me of one of the very first clever maths thoughts I remember having, during my Cambridge entrance interview (I didn’t get in). The question was, “what’s the last digit of \(2004^{2004}\)?”, and the insight was that I don’t care about any of the other digits. I left the interview pleased with myself…