12/07/2025
"Mathematics may not be ready for such Problems."
----------Paul Erdős...famously said about the Collatz conjecture
C𝗼𝗹𝗹𝗮𝘁𝘇 𝗖𝗼𝗻𝗷𝗲𝗰𝘁𝘂𝗿𝗲: 𝗧𝗵𝗲 𝗘𝗮𝘀𝗶𝗲𝘀𝘁 𝗨𝗻𝘀𝗼𝗹𝘃𝗲𝗱 𝗠𝘆𝘀𝘁𝗲𝗿𝘆 𝗶𝗻 𝗠𝗮𝘁𝗵𝗲𝗺𝗮𝘁𝗶𝗰𝘀
The Collatz Conjecture, or the Kakutani Problem, or simply, 3n +1, is commonly known as the world’s most “infamous” Mathematical problem. Why infamous? Well, that’s because, since being created by Lothar Collatz in 1937, till date, no one has been able to solve this problem. Some even say it was invented back then by the Soviets to slow down American science, and it did a pretty good job at that! So, what is the Collatz Conjecture, really? Let’s see how it works!
The Collatz Conjecture is very easy to calculate. For example, let’s start with a random natural number from 1 to 10, say 3. Now, what we do is if the number is odd, we multiply it by 3 and add 1, so if the odd number is n, we simply apply 3n +1. So, if we apply that to 3, we get 10. Then, if we get an even number, we simply divide it by 2. So, 10 becomes 5. 5 is again odd, so we apply 3n +1 and get 16. It’s even, so we divide by 2 to get 8. Then we again halve it, since it’s even, and get 4. It’s again even, so we halve it again to get 2. Halving one last time, we get 1. But do we stop here? No, 1 is odd. So, we again apply 3n +1, but we get 4. Halving twice, we once more get 1. So, at this point, we realize we’re in sort of a loop. No matter how long we apply the formula, we still end up in a loop of 4 to 2 to 1 to 4 again. You may think, “Well, then I’ll just start again with another number, maybe 7 or 8. Surely, there’s at least one number which doesn’t end up in the loop, right?” Sadly, we don’t know if that is right or wrong.
The problem in Collatz Conjecture is to prove or disprove the theory that in this conjecture, every single number ends up in the same 4, 2, 1 loop. Till date, numerous Mathematicians have tried to prove this theory. And for that purpose, they have calculated the numbers through sheer brute force upto 2.95x1020, and although all numbers until that have similarly ended up in the 4, 2, 1 loop, no one has yet found a proof to say for sure that all natural numbers upto infinity abides by the law. So, some Mathematicians have started researching to disprove the theory by either (1) finding an anomaly in the conjecture, where one number grows infinitely instead of ending up in the 4, 2, 1 loop, or (2) finding a separate loop of numbers where calculating one number after another continuously constantly leads upto a same previous result. But till date, none of these counterexamples have been found either, not for lack of trying though.
But why does the result ultimately decrease to the 4, 2, 1 loop instead of continuously growing. From bird’s eye view, we see that the odd numbers are more than tripled, while the even numbers are only halved. The reason behind that is, when we halve an even number, it may result to either an odd or even number. But if we apply 3n +1 to an odd number, it is always going to become an even number! So,
