Observations
It is obvious from the table of PPDIs
that PPDIs are not exactly rare. The number of PPDIs seems generally to increase
with base, and the order of the largest PPDI seems generally to increase with
base. Specifically, it appears that (1) all bases contain PPDIs, and (2) it may
be possible to bound tightly the highest order PPDI as a function of base.
Observation (1) is treated below. Those interested in examining observation (2)
should study the article in Journal
of Recreational Mathematics referred to elsewhere. HPDIs are not rare
either, there being between 0 and 5 in bases 2-10. Theorems
It is possible to prove some properties about the distribution of PPDIs. In
what follows, the notation { f } refers to the numeral whose value
is f. (The braces may be omitted where no ambiguity results.) In what
follows, we do not necessarily distinguish between a number (i.e., an abstract
value) and its representation in a given base. In all cases, the meaning should
be clear. Theorem 1. Every odd
base (3, 5, … ) contains at least one nontrivial (multi-digit) PPDI. Proof.
It is easy to show that { k+1 }{ k+1 }2k+1,
where k = 1, 2, … is a PPDI. In particular, in base-(2k+1),
{ k+1 }{ k+1 } represents the value (k+1)(2k+1) +
(k+1) or 2k2 + 4k + 2. But this is
just (k+1)2 + (k+1)2, which, by
definition, makes { k+1 }{ k+1 } a PPDI in
base-(2k+1). Thus, 22 is a base-3 PPDI, 33 is a base-5 PPDI, etc.
Q.E.D In general, such
existence theorems are proved by showing that the positional value and summation
value of numbers in a particular form are equal. The details are tedious and are
omitted in the proofs that follow. Theorem
2. There is at least one nontrivial PPDI in every base b = 3k+1,
where k = 1, 2, … . Proof. { k }{ 2k+1 }{ 0 }b is a
PPDI. Thus 130 is a base-4 PPDI, { 5 }{ 11 }{ 0 } is a base-16 PPDI, etc.
Q.E.D In light of Theorem 1, Theorem 2 is of greatest interest
for odd values of k, which involve even bases. In a similar fashion, this
next theorem is most interesting for even values of k, which address even
bases. Theorem 3.
There is at least one nontrivial PPDI in every base b = 3k+2,
where k = 1, 2, … . Proof. { k }{ 0 }{ 2k+1 }b is a
PPDI. Thus, 103 is a base-5 PPDI, 205 is a base-8 PPDI,
etc. Q.E.D Theorem 4. There is at least
one nontrivial PPDI in every base b = 18k+6, where k = 0, 1,
… . Proof. { 14k+5 }{ 4k+1 }{ 12k+4 }b is a
PPDI. Thus, 514 is a base-6 PPDI,
and
{ 19 }{ 5 }{ 16 } is a
base-24 PPDI, etc. Q.E.D Theorem 5.
There is at least one nontrivial PPDI in every base b = 18k+12,
where k = 0, 1, … . Proof. { 10k+6 }{ 8k+6 }{ 12k+8 }b is a
PPDI. Thus, 668 is a base-12 PPDI,
and
{ 16 }{ 14 }{ 20 } is a base-30 PPDI, etc.
Q.E.D Theorem
6. There is at least one nontrivial PPDI in every base b = 18k2,
where k = 1, 2, … .
Proof. { 9k2−3k }{ 9k2 }b is a
PPDI. Thus, 69 is a base-18 PPDI, and { 30 }{ 36 } is a
base-72 PPDI, etc.
Q.E.D Theorem
7. There are infinitely many numbers that are nontrivial PPDIs in at
least two bases. Proof. { k }{ 2k+1 }{ 0 }3k+1 = { k }{ 0 }{ 2k+1 }3k+2,
where k = 1, 2, … . This follows directly from theorems 2 and 3. For
example, 1304 and 1035 are both PPDIs, and 28 = 1304 = 1035. Of course, no fixed string of numerals—except length-1
strings—can be a PPDI in two different bases, as the sum of the powers of the
digits is
characteristic of the numeral string itself.
Q.E.D We now consider the
matter of the distribution of PDIs raised earlier. We
begin by noting that all bases contain trivial PDIs. In particular, { 0 }
is a PDI in every base b and every order k. The same is true for{
1 }. These results follow immediately from the PDI definition. Also, where 1 <
i < b, { i }b is trivially an order-1 PDI.
It is not known if there are
bases lacking nontrivial PDIs, and it is not known if there are bases containing
only a finite number of PDIs. Theorem 8, however, establishes that there are
bases with an infinite number of PDIs. Theorem
8. There are infinitely many bases in which there are infinitely many
nontrivial PDIs. Proof. Consider
the (j+1)-digit representation r = { i }{ 0 } …
{ 0 } in base i n, where i > 1, j > 0,
and n > 1. (Thus, r is i followed by j zeroes.)
Thus, from our definitions, we have
P(r) = i (i n) j = i n j + 1 = Sn j + 1 (r),
that is, r is a base-i n order-(n j + 1) PDI. The
theorem tells us,
for example, that 204 is an order-3 PDI, 30027 is an order-7 PDI, etc.
Q.E.D Finally,
Michael and I were asked if there are any prime PPDIs in base-10. It isn't clear
to my why this should be an interesting question—primes are easily identified
in smaller bases in the table of PPDIs—but, for those who are interested, I note
that the 14-digit PPDI 28,116,440,335,967 is prime. |
