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I've been nerdsniped haha. Just found this quora question which is the same as what you asked: <https://www.quora.com/Assume-n-1-How-can-I-prove-one-can-onl...> But the answers seem wrong? The first one uses only the primes 2 to 7 instead of 2 to 13, but there's a reason I used 2 to 13, and it's that there's a counterexample if you use less than that. Even using 2 to 11, which is more than they use, a counterexample exists between 520 and 620, where there are 25 numbers which are not divisible by any of 2, 3, 5, 7, or 11: [521, 523, 527, 529, 533, 541, 547, 551, 557, 559, 563, 569, 571, 577, 583, 587, 589, 593, 599, 601, 607, 611, 613, 617, 619] For the proof by induction at the bottom of the page, I'll admit I just don't understand it, but at first glance it seems impossible to me that it would work given how short it is and it doing things that just don't seem to make sense. But maybe I'm missing something. Regardless, having triple-checked it, I will take pride that my proof is probably correct :P Edit: Now by double-checking this particular comment I've found that there are actually only 24 numbers between 520 and 620 which are not divisible by any prime from 2 to 11. 25 is if you use 2 to 7. This has no effect on the proof though, only this comment about the quora proofs, and even then my point stands. |