I can’t believe that I’m the first to follow this train of thought, since each step is obvious. Therefore, I am tagging this as a reference request. Someone please quote me chapter and verse where this thread has been dealt with before. Thanks.
Step 1. Suppose x > 1. Hmm, x might be really big, but we want to get a handle on x, to downsize x, without losing, so to speak, the essential properties of x. How can we do this?
Step 2. The Harmonic Sequence leaps to mind. Because of the divergence of the corresponding series (i.e., the Harmonic Series), if we start subtracting from x the terms of the Harmonic Sequence, it will be after only finitely many such subtractions that we exhaust x, not matter how big x was.
Step 3. Define H(n) to be the sum of the first n terms of the Harmonic Sequence, for each positive integer n. So, H(n) = 1/1 + 1/2 + … + 1/n.
Step 4. So, there is a unique positive integer n such that H(n) ≤ x < H(n + 1). Let us call that positive integer n the index of x, denoting it by index(x), and let us call the quantity x – H(n) the residue of x, denoting it by r(x).
Commentary. The residue of x might already by a useful quantity. Whether it is in fact so is what the title of this post is asking.
For example, is there a useful characterization of r(xy) in terms of r(x) and r(y), somewhat along the lines of characterizing the logarithm? Perhaps. Or perhaps to get non-trivial results, we must, as so often happens, enter into iteration. So, how can we continue the process, in a way that ties back to x in an essential way? Again, the answer is obvious.
Step 5. Consider the quantity H(n + 1) – x, where n is the index of x. Let’s call this the complementary residue of x, denoting it by c(x). It is obvious that 0 < c(x) <= H(n + 1). How can we best use c(x)? Again, the answer is obvious: take its reciprocal.
Step 6. For x > 1, define the sphinx of x to be 1/c(x), denoting it by sphinx(x). Note that for really large x, the sphinx of x will also be really large.
Step 7. For a given x > 1, set up the following 5-fold sequence:
pathx(1) = x
pathindex(1) = index(pathx(1))
pathr(1) = r(pathx(1))
pathc(1) = c(pathx(1))
pathsphinx(1) = 1/pathc(1)
For each positive integer n > 1:
pathx(n) = pathsphinx(n – 1)
pathindex(n) = index(pathx(n))
pathr(n) = r(pathx(n))
pathc(n) = c(pathx(n))
pathsphinx(n) = 1/pathc(n)
Step 8. For x > 1, define the distillation sequence of x to be the sequence {y(n)} of positive integers such that for each positive integer n, y(n) = pathindex(n).
The obvious question is then whethere, for a given x > 1, the distillation sequence of x characterizes x in any interesting or useful way. For example, x is algebraic fif its distillation sequence has such-and-such a property.
Regards,
Mike Jones
28.May.2011 (Beijing time)
Added by OP on May 30, 2011 (Beijing time)
My motivation for the original investigation along these lines was to solve the open problem of finding an explicit well-ordering for the set of real numbers. My strategy was simply that I would associate a unique sequence of positive integers with each real number, and then let the well-ordering of the set of positive integers induce a well-ordering on the set of real numbers. Since finding an explicit well-ordering for any non-empty connected subset of the reals is equivalent to finding an explicit well-ordering for the reals, it seemed apropos to deal with (1,infinity). So, this was the motivation for step 1, and the harvesting, so to speak, of the distillation sequence. However, it seems to me that this sequence is not unique for a given x > 1. However, the infinite matrix whose first row is the distillation sequence for x, whose second row is the distillation sequence for x + 1, and whose third row is the distillation sequence for x + 2, and so on MIGHT be uniquely defined by x. I call this matrix the Serengeti matrix for x. But we need a single unique sequence for x. Using the diagonal sequence of course suggests itself. I call this the Serengeti sequence for x. If the Serengeti sequence is unique to x, then that solves the problem, but I have not had the time/ability to determine this. But I thought that the partial results along the way, such as the distillation sequence possibly yielding useful information was of interest in its own right, but since high-rep users are balking at step 1, I guess I need to give a full account of my motivation, and so now you have it:-)