## Monday, September 21, 2020

### Root words (revisited)

A few days ago, Futility Closet highlighted a David Morice "Kickshaws" bit (from a 1997 "Word Ways") dealing with "root words" wherein the number of letters of a specific power of an integer written in English words is equal to that integer. This looked like something I could verify and possibly expand on. In short order, Mathematica came up with over one hundred examples. I trust of course that Mathematica is correctly providing the (American) English equivalents of the large numbers. Also, Mathematica does not ever include the word "and" in its integer wording. The solutions list begins:

3^0 = 1: one
4^1 = 4: four
9^2 = 81: eighty-one
10^2 = 100: one hundred
10^6 = 1000000: one million
10^7 = 10000000: ten million
10^9 = 1000000000: one billion
10^10 = 10000000000: ten billion
34^3 = 39304: thirty-nine thousand, three hundred four
50^7 = 781250000000: seven hundred eighty-one billion, two hundred fifty million
57^5 = 601692057: six hundred one million, six hundred ninety-two thousand, fifty-seven
60^9 = 10077696000000000: ten quadrillion, seventy-seven trillion, six hundred ninety-six billion
66^4 = 18974736: eighteen million, nine hundred seventy-four thousand, seven hundred thirty-six
70^8 = 576480100000000: five hundred seventy-six trillion, four hundred eighty billion, one hundred million
82^6 = 304006671424: three hundred four billion, six million, six hundred seventy-one thousand, four hundred twenty-four
84^5 = 4182119424: four billion, one hundred eighty-two million, one hundred nineteen thousand, four hundred twenty-four
88^5 = 5277319168: five billion, two hundred seventy-seven million, three hundred nineteen thousand, one hundred sixty-eight
88^6 = 464404086784: four hundred sixty-four billion, four hundred four million, eighty-six thousand, seven hundred eighty-four

In addition to the five powers of ten, 88 has both a fifth and a sixth power that qualify. Another integer with multiple powers (59 and 60) is 2326. My final entry (the full list is here):