Smallest Integer using Three Words in English Description
Theorem
The smallest integer which uses exactly $3$ words in its standard (British) English description is:
- $21 \, 000$: twenty-one thousand
counting hyphenations as separate words.
Proof
All integers up to $100$ (one hundred) use either $1$ or $2$ words:
- one
- sixty
- seventeen
- ninety-eight
All integers of the form $100 n$ for $n = 1, 2, \ldots 9$ use exactly $2$ words:
- one hundred
- seven hundred
- nine hundred
In British English, the technique for describing integers from $101$ to $199$, and $201$ to $299$ and so on, is to use and between the number of hundreds and the rest:
- one hundred and one
- three hundred and thirteen
- four hundred and twenty-six
- seven hundred and seventy
thus using either $4$ or $5$ words.
All integers of the form $1000 n$ for $n = 1, 2, \ldots 10$ use exactly $2$ words:
- two thousand
- five thousand
- eight thousand
- twelve thousand
- nineteen thousand
- twenty thousand
Similarly with hundreds, the technique for describing integers of the form $1000 m + n$ for $1 \le n \le 99$ is to use and between the number of thousands and the rest:
- five thousand and eighteen
- sixteen thousand and forty-eight
- thirty-seven thousand and sixty
thus using either $4$ or $5$ words.
All other numbers between $1100$ and $20 \, 999$ trivially use more than $3$ words:
- four thousand, eight hundred
- sixteen thousand, one hundred and seventy-seven
- twenty thousand, nine hundred and ninety-nine
and so on.
The smallest integer to use exactly $2$ words is $21$:
- twenty-one
Hence:
- twenty-one thousand
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $21,000$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $21,000$