Infinite words containing the minimal number of repetitions

A square in a word is composed of two adjacent occurrences of a nonempty word. This note gives a simple proof and a straight construction of the existence of an infinite binary word that contains only three squares. No infinite binary word can contain fewer squares. The only factors of exponent larger than two that our infinite binary word contains are two cubes. Furthermore, we provide two additional results on alphabets of size 3 and 4. We prove that there exists an infinite overlap-free ternary word containing only one square. On a 4-letter alphabet we show there exists an infinite 3/2+3/2+-free 4-ary word containing only one 3/2-power.