I said simple, not easy!
The Collatz Conjecture, also known as the 3n+1 problem, has been driving mathematicians mad since the mid 1930s when it was it was first proposed by Lothar Collatz. Although it is simple to understand — as it involves nothing more than addition and division — it is not simple to prove.
Here’s the problem:
Take a given integer n. If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Continue the pattern taking the answer as your new n.
Let’s look at some examples:
3, 10, 5, 16, 8, 4, 2, 1…
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1…
12, 6, 3, 10, 5, 16, 8, 4, 2, 1…
21, 64, 32, 16, 8, 4, 2, 1…
You will notice that for each of these numbers, you end up at 1, which would lead to an infinite loop of 1, 4, 2, 1, 4, 2, 1…
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But is that the case for all integers? That’s the Collatz conjecture, and it has never been proven. It has been tested for absurdly large numbers — for example, the largest number of steps it takes for a number below a billion to reach 1 is 986 — but nobody has been able to say for sure that no number exists that won’t ever reach 1.
H. S. M. Coxeter wrote of the difficulty of finding a counter-proof to the conjecture: “I must warn you not to try this in your heads or on the back of an old envelope, because the result has been tested with an electronic computer for all x1 ≤ 500,000. This means that, if the conjecture is false, the prizewinner muse [sic] either find a sequence of this kind which he can prove to be divergent, or else find a cyclic sequence for this kind whose terms are all greater than half a million.”
But, if you are still insistent, there is some prize money involved. Paul Erdos said of the conjecture: "Mathematics is not yet ready for such problems," but he offered $500 for a solution. Thwaites, after whom the problem is sometimes called the Thwaites’ conjecture, offered up £1000 (about $1500) as well.
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