h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I submit the abstract of a paper which may be viewed in its entirety at www-personal.ksu.edu/~kconrow/newpapr2.pdf. I would appreciate
either positive or negative comments.

Several major steps develop a proof of the Collatz conjecture. (1)The overall structure of the Collatz graph is a binary predecessor tree whose nodes contain residue sets, $\{c[d]\}$. (2)The residue sets are deployed as extensions (right descents) and l.d.a.s (left descents). (3)Generation of a large initial sample of the tree was accomplished using a recursive computer program. (4)The program output was sorted to show that residue sets which head the l.d.a.s and share a common modulus (the $d$ of $\{c[d]\}$) have multiplicities precisely equal to the successive elements of the Fibonacci series. (5)The density of integers in residue sets provides a measure of the contents of each set, hence that of the whole tree by summation. (6)Use of the established infinite
summation, $\sum_1^\infty F_{i}/2^{i+1}= 1$, shows that all positive
integers are accounted for, thus effecting a proof of the Collatz
conjecture.

I suggest you submit your proof to a professional mathematician or to a mathematics journal. I don't think there is anyone here capable of judging it.

Thanks for your quick response. You do list number theory and mathematics among your interests. I'm trying to reach Sir Thwaites himself. Do you happen to know his email address? -- Ken

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A good experiment would be to modify your code so that it works for the 3n-1 problem. (This should be easy for you to do; if I remember correctly, you did some modulo 8 operations. The code for least-residues of 1 and 7 would be switched and the code for least-residues of 3 and 5 would be switched. Forgive me if I'm completely off base here; your code is hard for me to follow.) Then if you still get complete coverage, your code is flawed (due to there being cycles other than 4, 2, 1). If you detect the other two cycles, then that's another feather in your cap.

If I understand you correctly, you've devised a "covering" algorithm. Since as far as anyone can tell, the Collatz graph contains all of the natural numbers, a large number of covering algorithms could be devised, most of them having nothing to do with the Collatz problem. (The ultimate covering algorithm would be the Collatz graph itself and its supposed existence obviously doesn't constitute a proof). Then one could imagine covering algorithms having a bit more to do with the Collatz problem. How do you know when you've captured enough of the essence of the Collatz problem? It seems to me that devising a covering algorithm is not enough to constitute a proof. Consider Terras' proof that almost all natural numbers have finite stopping time. Since the results of your program require interpretation, how do you know that there isn't some number that doesn't have finite stopping time? Personally, I find it hard to believe that Fibonacci numbers have anything to do with the Collatz conjecture, but I've seen more bizarre things in my empirical investigations. Anyway, this is the best I can do without digging into the software.