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.

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 = 3³ + 4⁴ + 3³ + 5⁵ = 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 1¹ = 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 10₂.

Proof. It takes a moment to appreciate what is going on here. Of course, 1¹ = 1, but it is also true that 0⁰ = 1. This means that S^D(10₂) = 1¹ + 0⁰ = 2 = P₂(10₂). 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(100110₂) = 6, but 100110₂, that is, P₂(100110₂) = 38. This means that neither 0₂ nor 11₂ 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 10₂ 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. 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 0⁰ = 0, whereas, in discrete mathematics, 0⁰ is usually assigned the value 1. I am inclined to accept this latter convention, though I suppose that one can argue for 0⁰ = 0, since that would mean that zeroes do not contribute to the sums defined for PDDIs, just as zeroes do not contribute to the sums defined for PDIs and PPDIs.

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) = 3³ + 4² + 5¹ = 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 n¹ = 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, …, d₁). The rightmost term in P_b(R) is d₁ b⁰ = d₁. Likewise, the rightmost term in S^P(R) is d₁¹ = d₁. 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_ib^i-1. The corresponding term in S^P(R) is
d_iⁱ = d_id_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_ib^i-1 > d_id_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 Pb(R)	Nontrivial Examples
Armstrong number of the first kind (AN1)	Pluperfect digital invariant (PPDI) of order m	Sm(R)	Yes
Armstrong number of the second kind (AN2)	—	SP(R)	No
Armstrong number of the third kind (AN3)	Perfect digit-to-digit invariant (PDDI) or Munchausen number	SD(R)	Yes
Armstrong number of the fourth kind (AN4)	Perfect digital invariant (PDI) of order k	Sk(R)	Yes

Finally, I offer some research topics:

1. Find a tight upper bound for the order of the largest possible AN3 in base b.
2. Find all AN3s in bases above 10.
3. Characterize the bases for which there are no nontrivial AN3s.

As always, I invite questions and new results. Write me at the address below.

 
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