Since the thread is mostly jokes, here's roughly where the actual research stands.
Brute force: every starting number up to 2^71, about 2.4 x 10^21, has been checked and all of them fall into 4-2-1. That's Bařina's 2025 run.
A second loop is nearly ruled out too. Hercher (2023) showed there's no non-trivial cycle built from 91 or fewer up-then-down blocks, and the length floor for any cycle that could still exist is on the order of 100 billion terms. So a counterexample almost certainly isn't a small loop nobody noticed. It would be one trajectory that just keeps climbing forever.
Tao's 2019 paper is the closest anyone has gotten to ruling that out. He proved that almost all starting numbers (in logarithmic density) eventually drop below any function you name that goes to infinity, even something as slow as log log log log N. It doesn't cover every number, but it means a runaway trajectory, if one exists, is extremely rare.
Worth knowing why this is genuinely hard and not just unsolved: Conway proved in 1972 that the generalized version, same setup but other multipliers, is undecidable. There's no single algorithm that settles every problem of this shape. The plain 3x+1 case might still be provable, but that's the reason "just analyze the recursion" keeps stalling out.
So the computational angles are basically spent. What's left is number theory, and the 700k looks safe.
Since the thread is mostly jokes, here's roughly where the actual research stands.
Brute force: every starting number up to 2^71, about 2.4 x 10^21, has been checked and all of them fall into 4-2-1. That's Bařina's 2025 run.
A second loop is nearly ruled out too. Hercher (2023) showed there's no non-trivial cycle built from 91 or fewer up-then-down blocks, and the length floor for any cycle that could still exist is on the order of 100 billion terms. So a counterexample almost certainly isn't a small loop nobody noticed. It would be one trajectory that just keeps climbing forever.
Tao's 2019 paper is the closest anyone has gotten to ruling that out. He proved that almost all starting numbers (in logarithmic density) eventually drop below any function you name that goes to infinity, even something as slow as log log log log N. It doesn't cover every number, but it means a runaway trajectory, if one exists, is extremely rare.
Worth knowing why this is genuinely hard and not just unsolved: Conway proved in 1972 that the generalized version, same setup but other multipliers, is undecidable. There's no single algorithm that settles every problem of this shape. The plain 3x+1 case might still be provable, but that's the reason "just analyze the recursion" keeps stalling out.
So the computational angles are basically spent. What's left is number theory, and the 700k looks safe.