#math trivia for #August26: #238 has the form abc where a^b = c. Which other three-digit numbers have this form? (Some may not be obvious.)

— Burt Kaliski Jr. (@modulomathy) August 27, 2013

The other numbers are:

- 101, 111, 121, 131, 141, 151, 161, 171, 181, 191
- 201, 212, 224 (next would be the current 238)
- 301, 313, 329
- 401, 414
- 501, 515
- 601, 616
- 701, 717
- 801, 818
- 901, 919

The ones that may not be so obvious are the ones with *b* = 0. But recall that the 0th power of every number is 1. This makes the exponentiation law work out: *x*^{(y+z)} = *x*^{y} * *x*^{z}. If *z* = 0, then *x*^{(y+z)} = *x*^{y}, which requires that *x*^{z} be 1.

Other trivia: Most of these numbers could be US area codes — which ones couldn’t? And many could be route numbers of US Interstate highways — again, which ones couldn’t? Notably, 238 itself is one of the anomalies in the Internet highway numbering system, more than a trivial concern to highway buffs.