**Corollary
1.** There is at least one nontrivial PPDI in every base *b* >
2, except possibly where *b* = 18*k* and *k* is not a perfect
square.
Corollary 1 seems positively
goofy. Is there really something special about bases that are multiples of
18? Likely not, and the exceptions in the corollary have bothered me for
years. Recently, I decided to revisit the distribution of PPDIs. My conjecture,
of course, is that all bases above 2 contain nontrivial PPDIs. Even if my
conjecture is correct, however, proving it may be difficult. The foregoing proofs all
rely on two- and three-digit PPDIs, but PPDIs that might lead to arguments that
could plug
the ugly hole in Corollary 1 seem hard to come by. For example, Dik Winter of
Centrum voor Wiskunde en Informatica in Amsterdam recently discovered that the
smallest non-trivial PPDI in base-90 (i.e., base-18*k*, where *k *= 5)
is {73}{62}{15}{62}{83}{18}{39}{47}. (Dik Winter has a brief discussion of
“Armstrong numbers,” a.k.a. PPDIs, and two interesting tables of such
numbers. These can be found here.)
Of course, there might be an existence proof for the conjecture that does not
actually identify the numbers that make the conjecture true, but it is not at
all clear what sort of argument might be constructed. Alternatively, there may
actually *be* bases without multi-digit PPDIs, but it is equally unclear
how to approach proof of *this* conjecture.
Mike
Keith and I have begun to eliminate some of the restrictions of Corollary 1.
What follows has resulted from our collaboration: **Theorem
9.** There is at least one nontrivial PPDI in every base *b* =
36*k*^{2}, where *k *= 1, 2, ... .
**Proof.** { 12*k*^{2}-4*k *}{ 24*k*^{2}-2*k *}{ 1 }
is a PPDI in base *b* = 36*k*^{2}. Thus, {8}{22}{1} is a
base-36 PPDI, {40}{92}(1} is a base-144 PPDI, etc.
**Theorem
10.** There is at least one nontrivial PPDI in every base *b* =
90*k*+18, where *k *= 0, 1, ... .
**Proof.** { 18*k*+4* *}{ 36*k*+8* *}
is a PPDI in base *b* = 90*k*+18. Thus, {4}{8} is a base-18 PPDI,
{22}{44} is a base-108 PPDI, etc.
**Theorem
11.** There is at least one nontrivial PPDI in every base *b* =
90*k*+72, where *k *= 0, 1, ... .
**Proof.** { 18*k*+14* *}{ 36*k*+29* *}
is a PPDI in base *b* = 90*k*+72. Thus, {14}{29} is a base-72
PPDI, {32}{65} is a base-162 PPDI, etc.
Now we are ready to improve
upon Corollary 1. Using that result and those above, we can easily prove the
following: **Corollary 2.** There is at least one nontrivial PPDI in every base *b* >
2, except possibly where *b* = 18*k* and all the following are
true: (1) *k* is neither a perfect square nor twice a perfect square and
(2) neither *k*-1 nor *k*+1 is divisible by 5.
Notice that
we could prove that all bases above 2 have nontrivial PPDIs if we could show
that bases for which *k*, *k*+2, and *k*+3 are divisible by 5
have PPDIs. The *k* case seems especially difficult. We know that when *k* =
5 and *b* = 90, the smallest nontrivial PPDI is eight digits long.
Even assuming an eight-digit PPDI could be the basis of a theorem, finding and
proving an appropriate conjecture is likely to be difficult. In any case, Winter
has shown that no base between 2 and 1000 lacks nontrivial PPDIs. Finally,
Mike was able to prove another theorem that does little to close the existence
gap, but which is interesting for its own sake: **Theorem
11.** Let *t* be the *n*^{th} triangular number, *n* >
1. (The triangular numbers are those representing the number of objects needed
to form a triangle like that formed by bowling pins. The first triangular number
is 1, the second—reflecting a row of 1 object and a row of 2 objects—is 3,
etc. In the case of bowling, i.e., tenpins, there are four rows of 1, 2, 3, and
4 pins. The 4^{th} triangular number is 10.) If *t* is a square
triangular number, let its square root be *r*. Then number {*r*}{*r*}{0}
is a PPDI in base *n*.
**Proof.** {*r*}{*r*}{0}
represents the value *rn*^{2} *+ rn* = *r* (*n*^{2}
+ *n*). Since the formula for the *n*^{th} triangular number *t*
is (*n*^{2} + *n*)/2, however, and since, by
assumption *t* = *r*^{2}, we may write this as 2*r*^{3}.
But, of course, the sum of the cubes of the digits of {*r*}{*r*}{0} is
also 2*r*^{3}, so {*r*}{*r*}{0} is, by definition, a PPDI.
Unfortunately,
square triangular numbers are rather sparse. The first two PPDIs whose existence
is asserted by the theorem are 660_{8} and {35}{35}{0}_{49}. |