Gareth McCaughan <gareth.mccaughan@pobox.com> wrote [about my first table]:

I have been thinking half-heartedly (and apparently half-brainedly) about this one from time to time ... and in the process of writing down my incoherent thoughts I realise that I have solved it.

I often do the same. See https://en.wikipedia.org/wiki/Rubber_duck_debugging

The value in row m, column n is the k such that m^k = 2n^k-1. For the (1,1) entry any k will do, hence the slightly curious "all" in that spot.

Right. My intention was for k to be the power necessary for the power mean of 1 and m to equal n. For those not familiar with the concept, the xth power mean of a and b is defined as (a^x + b^x)/2 = m^x. Common examples include: -1: Harmonic mean 0: Geometric mean 1: Arithmetic mean 2: Quadratic mean, aka Root Mean Square (RMS) But x can be any real number, not just an integer. For my first table I chose values of x that made the power mean of 1 and an integer (the row number) another integer (the column number). Most such values of x are transcendental numbers. But not all. What rational values can x have? Can it have an irrational algebraic value? (I'm speaking of the row and column both being positive integers, and the column number being less than the row number.) [About my second table:]

But it turns out that the entries in the table satisfy the following (defining) equation: if the entry in position (x,y) is z, then x^z + y^z = 2z^z.

Right.

It is not obvious to me whether this is actually equivalent to some sort of generalized mean as defined above. I'd guess not. What of course _is_ true is that z = ((x^z+y^z)/2)^1/z, so each entry is _a_ "power mean" of its row and column positions ... but with an exponent that equals the entry itself :-).

Yes, that was my intention. For every set of positive real numbers x<z1<y, z1 is some power mean of x and y. And for every positive real numbers x, y, and real number z2, there's a z2 power mean z1 that is between x and y. Since, given fixed x and y, z1 and z2 are monotonic functions of each other, it necessarily follows that for every x and y there must be a unique z1 that equals z2. I got curious what they looked like. 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?