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Is there some quality of the Collatz Conjecture that has made it so difficult to prove or disprove? Besides just using a computer to calculate lots and lots of values of $n$, of course.

Rodrigo de Azevedo
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JBraha
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    I think it's just that integers in general are tough to deal with (thanks to primes being a pain) – Rushabh Mehta Mar 30 '19 at 02:46
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    Voting to close as duplicate because the accepted answer explains it well: "I think the basic question being touched on is: In what ways does the prime factorization of $a$ affect the prime factorization of $a+1$?" –  Mar 30 '19 at 04:03
  • Perhaps it were a good idea to -somehow- combine this question/answers with other Q&A on the same subject: what are the mathematical reasons for the difficulty (instead of the "sociological" reasons - "...difficult because no one could solve it..." as mentioned in another answer below). This could become then a "big list" and "community wiki" – Gottfried Helms Mar 30 '19 at 08:13

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It is considered difficult because no one has been able to solve it.

The value of a problem like the Collatz conjecture isn't in the result. If the problem had been solved within a day of being proposed it might appear as an exercise somewhere.

It is precisely that its solution has eluded the finest mathematical minds that makes it an important problem. It means that, when someone finally solves it, he or she will have to think in new and creative ways. It is the hope that the attempts and the eventual solution will generate new mathematics that will be useful in solving other problems whose results are important.

John Douma
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  • I wasn't really asking why it was considered difficult, I was asking why it has eluded the greatest minds in mathematics. – JBraha Mar 30 '19 at 12:29
  • @JBraha Mathematics problems aren't classified as hard like search problems in Computer Science. You cannot pose a math problem and say I know this will be hard because it possesses certain properties. A math problem is considered hard precisely because many have failed to solve it. Perhaps you can research the history of the problem and see for yourself why the particular attempts have failed. – John Douma Mar 30 '19 at 15:37