"Quadratic surd" and "square root of a rational" are not the same thing. 1 + V2 is a quadratic surd, but its square is 3 + 2V2, which is not rational. On Sat, Feb 10, 2018 at 10:41 AM, Keith F. Lynch <kfl@keithlynch.net> wrote:
I've been discussing when the integer power mean of two positive integers is an integer. When it's *not* an integer, it's still obvious what kind of number it is. The arithmetic or harmonic mean of two integers is always a rational number. The quadratic (2) or citardauq (-2) power mean of two integers is always a quadratic surd, i.e. the square root of a rational number. The cubic (3) or cibuc power mean of two integers is always a the cube root of a rational number.
But what about the opposite question? Every number c between the positive integers a and b is the xth power mean of a and b for some number x. It's the unique non-zero solution to (a^x + b^x) / 2 = c^x. (Except when c is the geometric mean of a and b, in which case it *is* zero.) It's negative if c is less than the geometric mean of a and b, i.e. if c^2 < ab. Every x corresponds to a unique number between a and b, and it's monotonic, i.e. increasing x always increases the mean.
I'm interested in the cases where c is an integer. The first non-integer example of such an x is (1^x + 4^x) / 2 = 3^x, for which the solution is approximately 2.261405723477886. What kind of number is x, in the general case? Obviously it's sometimes an integer. When it's not an integer, is it ever rational? Is it ever algebraic, and if so, of what order? Is it ever transcendental? Is it in this case?
I eventually realized that the formula I found, c = ab/q, applies, not just to the quadratic and citardauq means, but to all power means where c is the xth mean and q is the -xth mean. That explains why the arithmetic-harmonic mean always equals the geometric mean. So do the quadratic-citardauq mean, the cubic-cibuc mean, etc.
The arithmetic-harmonic mean is what you get when you separately take the arithmetic mean of two number and the harmonic mean of the same two numbers, then you repeat the process with those two means, then you repeat the process with those two means or means, etc., always getting two new numbers, which rapidly approach each other. Analogously with the better known arithmetic-geometric mean.
All power means are, for lack of a better word, slideruleable. For any xth power mean, all numbers can be arranged in order on a line such that the xth power mean of a and b is always halfway between a and b. For instance for the arithmetic mean, the numbers are equally spaced, like on a ruler. For the geometric mean, a log scale is used. This is *not* the case with the arithmetic-geometric mean. Is there an official name for this useful property?
(The term "number" in the above should always be read as "real number." It's way too early in the morning (10:42) for me to think about complex numbers, especially complex exponents.)
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