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