In Defense of Base-Related Problems

In "A Mathematicians Apology" Godfrey Hardy remarked that questions
involving the decimal digits of numbers constitute non-serious
mathematics.  This has long been conventional wisdom.  For instance,
discussions of the decimal digits of PI very often elicit disdainful
comments pointing out that one might just as well consider the digits
of pi in some other base, and hence the problem is arbitrary and of 
no mathematical significance.  Just for fun, let me argue the case 
that this conventional wisdom is wrong.

First, I would suggest that some of the disdain for base-related
problems in mathematics is carried over, perhaps subliminally, from 
UNITS-dependent questions in physics, which are undeniably naive and 
pointless.  For example, there is a certain kind of person who will 
spend years studying the numerical properties of, say, the number 
186282, because he has heard that this is the Speed of Light - in 
units of miles per sec.  This is certainly misguided activity, but 
we should be clear about WHY it's misguided.  The person may actually
turn up some intersting and non-trivial mathematical properties 
related to the decimal number 186282, so if he were engaged in the
study purely for its mathematical content, the worst we could say is
that it's a rather highly specialized choice of subjects.   However, 
the study is flat-out misguided if the person is laboring under the 
impression that the numerical properties of the integer 186282 are 
related to the physical phenomenon of light - unless one is trying to 
make some anthropomorphic point about the significance of our choice
of units.

In contrast, consider a problem concerning the decimal digits of pi.
It's tempting to think that these digits are as unrelated to the
mathematics of PI as the digits 186282 are unrelated to the physics
of light propagation...but of course that's not true.  Although the 
choice of base 10 is not unique, the integer 10 is one of the smallest
integers, the first square-free composite number, etc., and the
decimal digits of pi represent mathematically meaningful information
about the relationship between the integer 10 and the transcendental
number pi.  For any given natural number B greater than 1 we have
the UNIQUE sequence of integers c0,c1,... in the range 0 to B-1 such 
that 
                pi = c0 + c1/B + c2/B^2 + c3/B^3 + ...

This is a perfectly respectable mathematical construction, and the
digits express a meaningful ANALYSIS of the transcendental number 
pi.  We can liken it to the Fourier expansion of a signal into its
constituient harmonics, or the expansion of a function into a sequence
of orthogonal functions.  It's true that we're free to select the
basis of our expansion (e.g., Legendre polynomials, simple sine 
functions, or any of infinitely many other possible bases), but this
does not imply that these expansions are mathematically meaningless.
Each possible well-defined expansion represents an analysis of the
source (e.g., the number pi) relative to a particular basis.

It is not necessarily obvious, apriori, what basis of analysis might
reveal some interesting mathematical structure.  For example, it has
recently been shown that an algorithm exists to compute the nth 
hexidecimal (base 16) digit of pi without computing the preceding
digits.  This particular construction does not work for arbitrary
bases, only for certain bases like 16.  It is not known whether any
analogous special attributes characterize the base-10 digits of pi,
but the point is that the base-B expansion of pi has actually turned
out to exhibit non-trivial structure for certain values of B.

In general, for any given B, the base-B representation defines an
extremely useful mapping between the reals and the set of integer 
sequences [c0,c1..] where the ci are restricted to the range 0 to B-1.
In fact, this mapping is by far the most widely used instrument
of mathematics for practical (not to mention impractical) purposes.
It's perfectly legitimate to ask if there are any interesting 
number-theoretic properties for certain specific sequences [c0,c1..].
For example, setting B=2, we know that numbers of the form  111..111 
(which are called Mersenne numbers) can be tested for primality (using
the Lucas-Lehmer test) much faster than the general case because of 
their special structure.  Thus, the real number 1.11111.... (base 2)
can be construed as a sequence of integers 1, 11, 111, ...  whose 
special  structure can be exploited for non-trivial number-theoretic
purposes.  Do the base B representations of any other real numbers 
(e.g., PI or e) yield sequences of integers with exploitable 
structure?

To be candid, another reason that base-related problems are usually 
avoided is that they tend to be extremely difficult.  The difficulty 
is due to the non-linearity imposed by the constraint that the
coefficients must all be in the range 0 to B-1.  This is what
distinguishes such questions from (and makes them much harder than)
questions about ordinary polynomials.  Consider the reverse-sum 
palidrome problem, i.e., beginning with a number 196, add the "reverse"
691, then iterate.  Will this necessarily lead to a palindrome (a 
number that equals its reverse)?  The answer seems to be no, but no
one can prove it.  There is a tendancy on the part of mathematicians,
when they can see no way of even approaching a problem, to declare
that the problem is simply insignificant and/or uninteresting.  For
example, recall Gauss' famous comment, when he was urged to tackle
Fermat's Last Theorem, that "Fermat's Theorem as an isolated 
proposition has very little interest for me, because I could easily
lay down a multitude of such propositions, which one could neither
prove nor dispose of".  Even today it it commonly said that Fermat's
Last Theorem, as an isolated proposition, is of little interest, but
the fact remains that important parts of modern number theory (e.g.,
Kummer's ideal theory) were developed specifically in efforts to solve
this "uninteresting problem".  It seems clear that the problem 
represents one particular consequence (and an easily stated one) of
some deep and profound mathematical facts, so it is disengenuous for
mathematicians to call it uninteresting or unimportant.  Does anyone
doubt that if Gauss had seen a way of proving this particular 
proposition, he would have done so?  He says the proposition has 
little interest for him, but this lack of interest is mainly due to
the fact that it is one of the (infintely many) propositions that
he feels he can neither prove nor disprove.  It's perfectly sensible
for people to choose to focus their attention on problems that they
believe they have some chance of solving, but we shouldn't make the
mistake of thinking that a problem is mathematically insignificant
simply because we have no clue how to solve it.

There is, however, validity in the argument that some problems are
"natural" whereas others are mere contrivances.  For example, it might
be argued that the "digital reversal" of a number is not a natural 
or meaningful concept, but consider a number N with the base-B
representation

         c0  +  c1 B  +  c2 B^2  +  ...  +  ck B^k

What are the "roots" of this number?  In other words, given the
coefficients c0,c1,...ck, for what values of B does the above
expression equal zero?  Now consider the digit reversal of N:

         ck + ... + c2 B^(k-2) + c1 B^(k-1) + c0 B^k

What are the roots of this number?  If we normalize these two 
polynomials the roots of N are simply the inverses of the roots of 
rev(N).  Thus, rev(N) can be expressed naturally in terms of the 
roots of N (and vice versa).

For other examples of non-trivial mathematical structure that
explicitly involves base-b representations of numbers, see the
notes
      Cyclic Divisibility
      Geometric Dot Products and Digit Reversals
      Powers of Primes Dividing Factorials
      Fibonacci, 1/89, And All That
      Generating Functions for Point Set Distances
      Least Significant Non-Zero Digit of n!

It's conceivable that all genuine mathematical structure has meaning 
and significance for reality.  In other words, it may be that there 
is no excess (unused) potential structure in the universe.  In any 
case, the structure of analytical expansions (including radix 
representations of real numbers) can certainly not be assumed to
be meaningless or insignificant.