Easy As ABC

In 2012, a Japanese mathematician named Shinichi Mochizuki announced that he finally knew the abc’s.

- Evelyn Lamb

He wasn’t talking about the Roman alphabet—he has known English for most of his life. He was talking about a mathematical statement called the abc conjecture. (In mathematics, a conjecture is a statement that some mathematicians believe to be true but no one has proved for certain. When it is proved for sure, it is called a theorem.)

Mochizuki studies number theory, a branch of mathematics. Mathematics is an incredibly broad, diverse subject. Mathematicians study everything from the random way stock markets and electrons seem to move to the properties of exotic shapes in unimaginable dimensions. But if you ask someone on the street what a mathematician does, the guess will probably be closest to what a number theorist does. They study the properties of numbers, especially the positive whole numbers: 1,2,3, and so on.

Connecting Multiplication and Addition

Number theorists do a lot more than count, though. They try to understand relationships between numbers. If you’ve taken algebra, you may have seen an equation that looks something like y =3 x + 4, or even y = mx + b. All those letters indicate variables. The expression y = 3x + 4 represents all the possible pairs of numbers that satisfy that relationship, which ends up looking like a straight line. If you substitute a number, like 2, for x in the first equation, you can find that y = 3 × 2 + 4, or 10.

The abc conjecture connects multiplication and addition: it relates two numbers and their sum to the factors of all three numbers. The conjecture starts innocently enough with what looks like one of the simplest possible equations: a + b = c. Here, the numbers a, b, and c all have to be positive whole numbers, and they can’t have any factors in common. For example, the equation 4 + 11 = 15 would be OK, but 4 + 12 = 16 would not be eligible because all three numbers have a common factor of 4.