Collatz conjecture

The Collatz conjecture is a conjecture (an idea that appears likely to be true but that has not been proven to be true) in mathematics. It is named after Lothar Collatz. He first proposed it in 1937, which was 2 years after getting his doctorate. It is about what happens when something is done repeatedly (over and over) starting at some positive integer n:

• If n is even (divisible by two), n is halved (divide by two = take its half).
• If n is odd (not divisible by two), n is changed to $3n+1$.

The conjecture states that if n is positive, n will always reach one and get stuck in the 4,2,1 loop as shown below. The problem is verified for all numbers below $2^{68}$ by brute force. This is still not proof that the conjecture is true because counterexamples (a number that disproves the conjecture) may be found in very large numbers. This has been shown in the disproven Pólya conjecture and Mertens conjecture.

Here is an example sequence:

• 9
• 28 (9 is odd, so we triple it and add one)
• 14 (28 is even; 14 is half of 28)
• 7 (14 is even, 7 is its half)
• 22 ($22=3\times 7+1$)
• 11
• 34
• 17
• 52
• 26
• 13
• 40
• 20
• 10
• 5
• 16 (16 is a power of two, so it will lead to 1, halving each time)
• 8
• 4
• 2
• 1 (after 1 comes 4, 2, 1, 4, 2, 1, etc.)

Debate

Many mathematicians argue over if it is really true. Numbers in the quadrillions have been tested but it has still remained true. Specifically, mathematicians have shown that a loop besides the 4,2,1 loop must be at least 186,000,000,000 (186 billion) numbers long. However, this is still very less compared to another conjecture that had been proved false in 1978. There are two outcomes where it is false: a number keeps growing towards infinity, or an extremely large number forms its own loop.