Consider the following information-theoretic argument. Suppose the four numbers A^n etc. have m bits in their binary representation. The equality relation among these numbers is a constraint on m bits (to leading order, as in an information rate). However, if A^n has m bits, then A has only m/5 bits. The number of bits we are allowed to search in finding an m-bit solution is therefore only 4(m/5). Okay, we can always be lucky. But barring a conspiracy, that becomes less and less probable as m increases. I’m not a number theorist, but I believe the ABC conjecture is about the non-existence of conspiracies of this kind. In the Swinnerton-Dyer type solutions, involving 6 integers, one is taking advantage of there being asymptotically more free bits than constraint bits (6(m/5) > m). -Veit
On Oct 10, 2020, at 9:59 AM, Dan Asimov <dasimov@earthlink.net> wrote:
1) What are the arguments for/against the existence of generalized taxicab numbers A^n + B^n = C^n + D^n ?
2) Is there a reason to believe that these should exist only for finitely many values of n ≥ 1 and positive A, B, C, D ???
3) And what if (for odd n) A, B, C, D are allowed to be negative???
—Dan