Daniel asked: which powers of 2 can you double without needing to carry digits? Clearly 1, 2, 4, 32, and 1024 all have this property, their doubles being 2, 4, 8, 64, and 2048 respectively. Are there any others?

Since I happen to have the powers of 2 up to 2^20 = 1,048,576 committed to memory since childhood, I confirmed that there were no other examples up to there: 128, 256, 512, 2048, etc. all require carries. So I told Daniel: I can’t find any other example, and on that basis, I conjecture that there aren’t any more. But if that conjecture is true, I don’t know if it will ever be proven, by humans or even AI!

Then I googled it, and saw that this is a known question (not very well known, but there’s a StackExchange post about it). And indeed it had been checked up to 2^millions, and no other counterexample had been found.

Why did I become confident so quickly that yes, 1024 is probably the last example of a power of 2 that can be doubled without carrying?

Because of the heuristic that the decimal digits of 2^n are more or less “random,” apart from various constraints that are irrelevant here (like that the last digit always cycles among 2, 4, 6, and 8). And 2^n has about n/log2 10 decimal digits. Since only 0, 1, 2, 3, 4 can be doubled without carrying, the probability of 2n being a counterexample should therefore be about (12)𝑛/log2 10.

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What all the above conjectures have in common, and what I find so fascinating about them, is that they seem hopeless to prove for exactly the same reason why they seem almost certainly true. Namely, they all seem to be true “merely” because it would be too insane of a coincidence were they false!

The trouble is, that’s not the sort of reason that seems amenable to turning into a proof. Fermat’s Last Theorem is an interesting exception that proves the rule here. That x^n+y^n=z^n has no nontrivial integer solution for n≥3, did seem almost certainly true on statistical grounds for n≥4 (and for the n=3 case, a proof goes back to Euler). And of course, FLT was ultimately provable. But Wiles’s eventual proof exploited a lucky connection between the Fermat equation and deep, fancy things like modular forms and elliptic curves. At no point did the proof formalize the statistical argument that a 12-year-old could understand, for why the theorem is “almost certainly true.” It simply had nothing to do with the statistical argument.