To my original introduction to digital invariants, I had
attached the following footnote to *pluperfect digital invariants*:
PPDIs are sometimes called *Armstrong numbers*, though I
have been unable to ascertain the source of this term. The term is
likely older than *pluperfect digital invariant*, though it
seems less useful. If you know the origin of the name *Armstrong
number*, I would like to hear from you.
This footnote attracted a few e-mail messages over the years, but
they were mostly inquiries as to whether I had yet discovered where the
term *Armstrong number* had come from. A note that arrived in my inbox on
May 1, 2010, was different, however. It came from one Michael F.
Armstrong, who made a credible claim to being the Armstrong of Armstrong
number fame. (See the post on my blog, “Mystery
Solved!”) In this section, I will discuss Armstrong and the
definitions he created nearly six decades ago.By his own account,
Armstrong created an assignment for a FORTRAN programming class at
the University of Rochester for which he defined what he called
*Armstrong numbers*. This occurred sometime around 1966. Armstrong is not
aware of being influenced by previous work, but only some of his
definitions seem to be original with him. Armstrong was a senior
systems programmer at the University of Rochester Computer Center and
sometimes taught programming classes. He believes he invented Armstrong
numbers for an Optics 209 course. My speculation that “Armstrong
number” is older than “perfect digital invariant” or “pluperfect digital
invariant” is probably wrong, but I don’t know for sure. It is still a
mystery how “Armstrong number” gained currency, as Armstrong himself
seems not to have had much to say about “his” numbers outside the
classroom. What I will do below is describe Armstrong numbers as they
were first defined and put them into a broader
perspective. At the end, I will offer some research problems. I should say at the outset that Armstrong was interested
primarily in computation. His definitions were intended to be
incorporated into search algorithms. His work as a mathematician ended
with his devising definitions. He and his students found some Armstrong
numbers, but they were hampered by the crude computers available at the
time. Armstrong defined Armstrong numbers
of the first, second, third, and fourth kind, a fact that has not found
its way into the literature. Moreover, his definitions assumed base-10
number representation, but they are easily generalized to other bases.
Armstrong’s original definitions can be found
here as a scanned image
of his “coffee-stained paper” from years ago. Rather than use his definitions directly,
I will restate them to be consistent in style with definitions
introduced earlier in my treatment of digital invariants. (See “Definitions.”
Readers should study this page carefully. I will be somewhat less formal
here by being less fussy about the distinction between a numeral/symbol
and the value it represents. This informality is unlikely to cause any real
confusion.) What
Armstrong described as an *Armstrong number of the first kind* is
simply a pluperfect digital invariant (PPDI) in base-10. I have defined
the *order *of the PPDI to be the number of digits it contains.
Armstrong made no such definition. References to “Armstrong
numbers” generally are to PPDIs in base-10, although it would be
reasonable to extend the definition to other bases. I will have more to
say about such numbers below. Armstrong *did* define *order*
for Armstrong numbers of the fourth kind, though, again, he did not
consider representations other than base-10. His *Armstrong number of the
fourth* kind is simply a perfect digital invariant (PDI), where the
order, as in my own definition of PDI, is the power to which the values
of individual digits are raised. (The use of “order” seems consistent,
as it refers to exponents both in the case of PDIs and PPDIs.) Particularly interesting is the
definition of *Armstrong number of the third kind*. In this case,
the conventional value of a representation is equal to the sum of the
values of its individual digits each raised to its own power. For
example, we have that 3435 = 3^{3}
+ 4^{4} + 3^{3} + 5^{5} = 27 + 256 + 27 + 3125 =
3435 I discovered this particular number; Armstrong did not exhibit an
Armstrong number of the third kind, which we can abbreviate as AN3.
Moreover, he speculated that there might not be any nontrivial AN3s. The smallest AN3 in base-10 (and any other base) is simply 1, though this
is a rather trivial example. To define Armstrong number of the third
kind more formally, let me introduce a definition.
**Definition.** Let the *
digit-determined summation
value* of a representation* R*, written *S*^{D}(R),
be
**Definition.** Let
*R* be the representation of a base-*b* number *n*. We
say that *n* is an *Armstrong number of the third kind in base-*b*
when* *P*_{b}(R) *= S*^{D}(R)
**Theorem 1.** In every base *b* =
*2*, 3, …, the
number 1 is an AN3.
**Proof.** This follows because 1^{1} = 1. We say that 1 is a trivial AN3.
*Q.E.D.*
**Theorem 2.** The only nontrivial
AN3s in base-2 is 2, that is 10_{2}.
**Proof.** It takes a moment to
appreciate what is going on here. Of course, 1^{1} = 1, but it is also true that 0^{0} = 1. This means that *S*^{D}(10_{2}*)* = 1^{1} + 0^{0} = 2 = *P*_{2}(10_{2}*)*. In fact, since each digit, whether 0 or 1,
contributes exactly 1 to the digit-defined summation value, *S*^{D}(R),
for
any binary representation *R*, *S*^{D}(R) is simply
*m*, where *m* is the length of representation *R*. For
example, *S*^{D}(100110_{2}*)* = 6, but 100110_{2},
that is, *P*_{2}(100110_{2}*)* = 38. This
means that neither 0_{2} nor 11_{2} is an AN3.
Now consider binary numbers at least three digits long (i.e., *m* >
2). The digit-defined summation value of each number is *m*. In
every case, however, the number (i.e., its positional value in base-2)
is greater than *m*. We prove this by induction. It is true when *m* = 3,
since the leading digit must be 1, which has the positional value of 4,
whereas the digit-defined summation value is only 3. Moreover, if it is
true for all binary representations of length *m*, it is true for
all representations of length *m*+1, which increases *S*^{D}
by 1 over that of every length-*m* representation. By assumption,
however, this is at least the value of the smallest positional value of
a length-*m* number. A length-(*m*+1) number, however, has a
positional value at least double that of a length-*m* number
because of the nature of positional number systems and the fact that the
leading digit is 1. In other words, there is no length-(*m*+1) AN3. Thus 10_{2} is the largest binary number
that is an AN3.
*Q.E.D.*
I’m not sure that last proof was very concise or very convincing, but,
after a little thought, the result is pretty obvious. This leads to a
more interesting result, for which I leave to the reader a more detailed
proof.
**Theorem 3.** In every base *b* =
*2*, 3, …, the
number of AN3s is finite.
**Proof.** For a representation of
length *m*, the largest *S*^{D}^{ } is *m* (b-1)^{b-1}.
This value increases geometrically with *m*. For a representation
of length *m*, the smallest *P*_{b} is b^{m-1}, which increases exponentially
with *m*.
Clearly, there exists some* n* such that, for all m > *n*,
*P*_{b} > *S*^{D}.
*Q.E.D.*
After achieving the results above, I discovered the brief paper “On
a curious property of 3435,” by Daan van Berkel. (Note that the
links in this paper to The On-Line Encyclopedia of Integer Sequences are
out of date. The Encyclopedia can now be found at
https://oeis.org/.) This 2009
contribution investigates AN3s under the name of *Munchausen numbers*.
The paper lists all the Munchausen numbers/AN3s in bases 2 through 10.
(There are but 21, including trivial ones.) It also offers a proof of
Theorem 3.
AN3s have also been
called *perfect digit-to-digit invariants* or PDDIs. See, for
example, the brief discussion by Harvey Heinz
here. Heinz identifies the number 438,579,088 as a PDDI, but this
requires that we define 0^{0} = 0, whereas, in discrete
mathematics, 0^{0} is usually assigned the value 1. Therefore, I
do not recognize 438,579,088 as a PDDI, and neither does van Berkel.
Finally, we get to *Armstrong numbers of the second kind*. For
this, we define yet another way of summing functions of individual
digits.
**Definition.** Let the *position-defined summation value* of a
representation *R*, written *S*^{P}(R), be
For example, *S*^{P}(345) = 3^{3} + 4^{2}
+ 5^{1} = 27 + 16 + 5 = 48.
**Definition.** Let
*R* be the representation of a base-*b* number *n*. We
say that *n* is an *Armstrong number of the second kind in base-*b*
whenever* *P*_{b}(R) *= S*^{P}(R)
We
abbreviate Armstrong number of the second kind as AN2.
Two theorems will
tell us all we need to know about AN2s.
**Theorem 4.** In every base *b* =
*2*, 3, …, each
number* n* = 0, 1, …,* b*-1 is an AN2.
**Proof.** The result follows
immediately from the fact that *n*^{1} = *n* for
each value of *n*, including 0.
*Q.E.D.*
There are thus trivial AN2s in every base. The next theorem, however,
demonstrates that there are no other AN2s. Armstrong speculated that
this might be the case.
**Theorem 5.** The is no nontrivial AN2
in any base.
**Proof.** The trivial AN2s all are
represented by a
single digit. We consider, therefore, representations having *m*
digits, where *m* > 1. If such a number *n* is to
be an AN2 in base *b*, we must have *P*_{b}(R) *= S*^{P}(R),
where *R* is the base-*b representation* of *n*.
Notice that both *P*_{b}(R) and *S*^{P}(R) are
the sums of *m* terms, each computed from one of the digits of *R*
(i.e., *d*_{m}, *d*_{m-1}, …, *d*_{1}).
The rightmost term in *P*_{b}(R) is *d*_{1} *
b*^{0} = *d*_{1}. Likewise, the rightmost
term in *S*^{P}(R) is *d*_{1}^{1} = *d*_{1}.
That is, the rightmost terms are equal.
Let* i* be an integer such that* m* ≥ *i* > 1. Consider the
corresponding terms in the two sums related to *d*_{i}. In *P*_{b}(R), we
have *d*_{i}b^{i}^{-1}. The corresponding
term in *S*^{P}(R) is d_{i}^{i} =
*d*_{i}d_{i}^{i}^{-1}. Since
*d*_{i} is a digit in a base-*b* representation of *
n*, it is necessarily the case that
*d*_{i} < *b*. Therefore, we must have that *d*_{i}b^{i}^{-1} >
*d*_{i}d_{i}^{i}^{-1}. This means
that, except for the rightmost term in the sums, all the terms of *P*_{b}(R) are strictly greater than the corresponding terms of *S*^{P}(R).
Since both sums have at least two terms, we must have that *P*_{b}(R) *> S*^{P}(R),
contrary to assumption.
*Q.E.D.*
In other words, Armstrong numbers of the second kind turn out to be not
very interesting.
It is now time to summarize Mike Armstrong’s definitions and to relate
them to other definitions. In the table below, *R* is an* m*-digit*
*representation of a number *n* in base *b.*
**Armstrong**
Name |
**Conventional**
Name |
**Value Equal**
to *P*_{b}(R) |
**Nontrivial **
Examples |
Armstrong number of
the first kind (AN1) |
Pluperfect digital invariant (PPDI)
of order *m* |
*S*^{m}(R) |
Yes |
Armstrong number of
the second kind (AN2) |
— |
*S*^{P}(R) |
No |
Armstrong number of
the third kind (AN3) |
Perfect digit-to-digit invariant
(PDDI) or Munchausen number |
*S*^{D}(R) |
Yes |
Armstrong number of
the fourth kind (AN4) |
Perfect digital invariant (PDI)
of order *k* |
*S*^{k}(R) |
Yes |
Finally, I offer some research topics:
- Find a tight upper bound for the order of the largest possible
AN3 in base
*b*.
- Find all AN3s in bases above 10.
- Characterize the bases for which there are no nontrivial AN3s.
As always, I invite questions and new results. Write me at the
address below. |