#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) = xy * xz. If z = 0, then x(y+z) = xy, which requires that xz 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.
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