**Why is it important?**

In the early 1990s, a team of scientists at MIT announced the solution to the problem known as the Millennium Prize. They set up a website that explained the problem and invited the world to compete. The contest attracted entries from many different countries and universities. The prize, worth $1 million, went to the team from Japan. But the Japanese team did not actually solve the problem—they only used a computer program that had been developed by a student at the University of California at Berkeley, who had worked on the problem for several years.

**What is this problem?**

The Millennium Prize is an example of a famous mathematical problem called the Collatz Conjecture. It was posed by a German mathematician named Lothar Collatz in the 1930s. The conjecture states that, for any positive integer x, the following recursive formula holds:

x = 2x + 1

If we start with a given positive integer, say x=10, we get the following sequence of numbers:

10 = 2 · 10 + 1 = 20 = 4 · 10 + 2 = 42 = 6 · 10 + 4 = 82 = 8 · 10 + 2 = 100 = 10 · 10 + 0 = 1000

**What’s going on here?**

The Collatz Conjecture is based on the idea that every positive number can be expressed as a unique product of powers of two and one. For example, the number 11, which is written as 11 = 101, is the product of two ones and one one:

11 = 101 = 2 · 10 + 1

For the first ten values of x, the first ten terms in the sequence above are:

10 = 2 · 10 + 1 = 20 = 4 · 10 + 2 = 42 = 6 · 10 + 4 = 82 = 8 · 10 + 2 = 100 = 10 · 10 + 0 = 1000

But when we reach the end of the sequence, the value of x changes, so we end up with an extra one in the final term. This is the crucial part of the conjecture: The last term in the sequence is always one more than the product of the previous two terms.

**Why does the conjecture matter?**

The Collatz Conjecture is the simplest, most general version of a much broader class of problems. Every positive integer x can be represented as the product of powers of 2 and 1, but the order in which the factors are multiplied doesn’t matter. For example, the number 17 can be written as the product of three twos and two ones:

17 = 12 · 52 + 2 · 11 = 101 = 102 · 22 + 1

There are other ways to represent x, and it turns out that the Collatz Conjecture is the only one that satisfies a property called the Collatz Conjecture. In other words, the conjecture says that, if we take any other expression of x as the product of powers of 2 and 1, then the first term in the resulting sequence must have a factor of one more than the previous one. If we write x = y2z, where y, z are positive integers, the conjecture becomes:

y2z = 2yz + 1

That is, the last term in the sequence must be one more than the product of the previous two terms.

**Why do we care about the Collatz Conjecture?**

The Collatz Conjecture has a wide range of applications. For example, it’s the basis for the world-famous Collatz conjecture, which was invented by the mathematician Lothar Collatz in 1936. The conjecture states that, for any positive integer x, the following recursive formula holds:

x = 2x + 1

If we start with a given positive integer, say x=10, we get the following sequence of numbers:

10 = 2 · 10 + 1 = 20 = 4 · 10 + 2 = 42 = 6 · 10 + 4 = 82 = 8 · 10 + 2 = 100 = 10 · 10 + 0 = 1000

In other words, if we let y=2x and z=x, then the Collatz Conjecture is equivalent to the statement that:

y2z = 2yz + 1

When we apply this to the Collatz Conjecture, we get:

2x = 2x + 1

This means that if we start with the number x, then the next number we get is two times the current one plus one:

2x = (2x) + 1

For example, if we start with x=10, we get:

10 = (2 · 10) + 1

= 20 = (4 · 10) + 2

= 42 = (6 · 10) + 4

= 82 = (8 · 10) + 2

= 100 = (10 · 10) + 0

And the last term is 100, not 101 as we might have expected.