Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
Keith F. Lynch wrote:
Most such values of x are transcendental numbers. But not all. What rational values can x have? Can it have an irrational algebraic value?
(This is values of x where m^x - 2n^x = 1, with m,n positive integers.)
Right, i.e. where n is the xth power mean of 1 and m. My first table.
I betcha x is never an irrational algebraic number, and I betcha there will not in our lifetimes be sufficient mathematical technology to prove this.
Maybe so, but I've made some progress on rational values of x. It's 0 whenever m is the square of n, 1 whenever m+1 is 2n, and 2 whenever m^2+1 = 2n^2, i.e. where m and n are corresponding elements of A002315 and A001653. For instance 7,5 and 41,29. There are infinitely many solutions for each of x=0, 1, and 2. It would also be m^x+1 = 2n^x for other integers x, but according to http://www.math.mcgill.ca/darmon/pub/Articles/Research/18.Merel/paper.pdf (equation 1) there are no such integers. It's 1/j whenever there are integers a and j such that m = a^j and n = ((a+1)/2)^j. For instance 9,4 for 1/2 and 27,8 for 1/3. There are infinitely many solutions for every positive integer j. It's 2/j whenever j is odd and m and n are corresponding jth powers of A002315 and A001653, for instance 343,125, i.e. 7^3, 5^3. There are infinitely many solutions for every positive integer j. (For even j, the solution for 2/j is that for 1/(j/2).) I can't find any other rational values of x. Can you?
Hence my second table. I assume they're all transcendental except of course where x=y, in which case z=x=y. Can anyone prove this? Or find a counterexample?
(This is values of x where m^x + n^x = 2x^x, with m,n positive integers.)
I betcha x is never algebraic unless m=n, and I betcha there will not in our lifetimes be sufficient mathematical technology to prove this.
If they are irrational, they are always transcendental. See the Gelfond-Schneider theorem, also known as Hilbert's 7th problem. That leaves the remote possibility that there's some non-integer rational x.