## Elliptic curves and the Collatz (3x+1) conjecture

I'm sure a lot of people are aware of the 3x+1 conjecture ( a quick recap: take a positive integer, strip powers of 2 off it until it is odd, multiply 3 and add 1. Repeat. Conjecturally, this always gives 1 after a finite number of iterations.) A lot of work has been done on this conjecture. Jeffrey Lagarias has a detailed survey article that is available on the Arxiv. Somehow, this problem seems not to have captured the imagination of the mathematical mainstream. Maybe everyone believes that such problems are intractable (or at least, as Erdos warned, beyond the current state-of-the-art.) I would like a suggest a highly speculative and half-baked approach which, if nothing else, betrays the fact I am obsessed with modularity and tend to look for it in places where it probably doesn't belong! Apologies in advance for what might be a waste of your time.

There are two ways the conjecture might be false:

1. Starting with an odd integer $x_0$, successive iterations produce a divergent sequence of odd integers $x_1$, $x_2$...
2. Starting with an odd integer $x_0 \neq 1$, we get back to $x_0$ after a finite number of iterations. (The sequence would then be $x_0, x_1, x_2, ..., x_n = x_0$ for some $n$.)

For each situation we construct the following family of elliptic curves:

$y^2 = x(x-x_i)(x + x_{i+1})$, $i = 0, 1, 2....$

Is it possible that either of the above situations will give us a family of elliptic curves with a member that is not modular? If at all this idea has potential, the second situation would be the one to consider initially since it will involve a finite number of elliptic curves.

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I think that the claim that "this problem seems not to have captured the imagination of the mathematical mainstream" is doubtful... I would imagine most mathematicians have spent non negligible time on the problem at some point, at the very least as students. – Mariano Suárez-Alvarez Apr 14 2010 at 3:48
In any case: do you have any reason to think there is any potential? For almost every sequence of integers one can consider such a sequence of elliptic curves and ask whether one is not modular... what has the Collatz problem that indicates this approach might work? Does your evidente also indicate that the same approach would work on proving there are not infinitely many Mersenne primes, say? – Mariano Suárez-Alvarez Apr 14 2010 at 3:53
There are other finite orbits if you allow negative $x_i$'s, such as {-1, -2} (maybe others, I cannot remember) so unless you propose a way to encode positivity of the $x_i$'s then that's yet another reason this cannot work. – BCnrd Apr 14 2010 at 4:33
Not a real question. – Daniel Moskovich Apr 14 2010 at 7:18
+1 for originality in seeking help from elliptic curves, rather than chaos or fractals. – Allen Knutson Apr 14 2010 at 14:14