All Numbers are Interesting

A Constructive Demonstration
Mike Keith, Aug 1998


 

Introduction

Most readers of this paper are undoubtedly familiar with the tongue-in-cheek "proof" that all numbers (usually taken to mean "positive integers") are interesting, which goes a follows: Classify all the positive integers into two sets: the interesting ones (the only even prime, the smallest Smith number, Hardy's taxicab number, etc.) and all the rest. Then we can find the smallest uninteresting number - which is surely interesting! - and, forthwith, transfer it to the interesting set. By continued iteration of this procedure, we eventually find that all numbers are interesting. (As a side comment, we note that even if we accept the tongue-in-cheek premises of this proof, it is only effective if the original set of uninteresting numbers is finite - a somewhat dubious assumption.)

In this paper, by contrast, we wish to show by direct construction that all positive integers are interesting, by exhibiting (or describing how to construct) a statement of the form

"n is the only positive integer with the property P(n)"

for every positive integer n. On the one hand, it is trivial to find a suitable property P - for example, we could say that 74 is the only positive integer whose prime decomposition is 2 x 37. But this is (to coin a phrase) uninteresting. What we want is a property P for which it is at least modestly unobvious that only n possesses the property.

Expository Numbers

Our basis for constructing a unique property of every positive integer is a generalization of G. H. Hardy's remark in A Mathematician's Apology [1]:

"There are just four numbers (after 1) which are the sums of the cubes of their digits, viz.

153 = 13 + 53 + 33, 370 = 33 + 73 + 03

371 = 33 + 73 + 13, 407 = 43 + 03 + 73

These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to a mathematician. The proofs are neither difficult nor interesting - merely a little tiresome."

The first generalization of this problem is to ask for integers that are equal to the sum of the kth power of their digits. We use the notation Sk(n) to stand for the sum of the kth powers of the decimal digits of n, so this asks for numbers for which

n = Sk(n)

In [2] this notion is further generalized to a sum of sums of powers of the digits of n; that is:

  t    
n =

Σ

  ci Ski(n )
  i=1

We now define a further generalization, in which the sum of the powers of digits of various powers of n are allowed:

  t            
n = Σ   ci Ski(n ai)       (1)
  i=1            

where the ci, ki, ai are all positive integers. If a set of parameters {ci, ki, ai}can be found such that equation (1) holds, then we call n an expository number (since its digits tell about itself).

Here are a few examples. The famous "number of the beast", 666, is an expository number, because

666 = S3(6662) + S1(6663)

That is, 666 is equal to the sum of the cubes of the digits in its square (6662 = 443556, and the sum of the cubes of these digits is 43 + 43 + 33 + 53 + 53 + 63 = 621) plus the sum of the digits in its cube (6663 = 295408296, and 2+9+5+4+0+8+2+9+6 = 45, and 621+45 = 666).

More dramatically, the integer 8112 is equal to three times the squares of the digits of its 23rd power:

8112 = 3 S2(811223)

We refer to an expository number with a given set of parameters {ci , ki , ai}as being of type c1k1(a1) + c2k2(a2) + ... + ctkt(at), with a ci being omitted if it equals 1. So 666 is of type 3(2) + 1(3) and 8112 is of type 32(23).

A fundamental fact about expository numbers is the following.

Theorem 1. There are only a finite number of expository numbers of any given type.

Proof: Let n have d decimal digits. We show that if d is sufficiently large, the maximum possible value of the RHS of (1) is larger than the minimum possible value of the LHS, and thus equality cannot hold. The maximum of the RHS is when all digits in each n ai equal 9. Since there are (at most) dai digits in n ai, the maximum possible value of Ski(nai) is dai9ki. On the other hand, the minimum possible value of the LHS is 10d-1. So equation (1) cannot be satisfied if

  t            
10d-1 > Σ   d ci ai 9ki = dK        
  i=1            

or

10d-1/d > K.

Since K is a constant and the left-hand side increases monotonically with d, then this inequality holds for all d sufficiently large (say, > D). This completes the proof.

(See [3] for a similar but more general theorem on the finiteness of the number of solutions of n = f(n), where f(n) is an arbitrary function of the digits of n.)

This proof also provides a lower bound on the size of integers that need to be checked when searching for expository numbers of a given type. We know there are none with D or more digits, and the maximum value that formula (1) can give for D-1 digits is (D-1)K, which we denote by L. Only the integers from 1 to L need to be checked.

Example: For type 3(2) + 1(3) discussed above, K = 293 + 391 = 1485. Thus the inequality holds when D=5 (since then 2000 > 1485), and the search limit is L = (D-1)K = 5940. Searching from 1 to 5940, we find that there are exactly two expository numbers of this type: 666 and 2583.

The Inverse Problem

As shown in the previous section, determination of all expository numbers of a given type is relatively straightforward, especially given the proliferation of fast computers and software packages that can handle numbers with many decimal digits. The inverse problem, however, is much more - dare we say it? - interesting: for a given integer n, is it an expository number at all? If so, of what types? We now describe an efficient method for tackling this problem. As we describe each step, we show the calculations for an example case, n = 74.

(Step 1): Compute all values of S1(na) that are < n. Then, for each value of a used there, compute Sk(na) for k=1, 2, 3, ... until Sk(na) > n. We now have all possible values of Sk(na) that are < n. For example, for n=74 there are ten k,a combinations with sum < 74:

k   a   Sk(74a)  
1   1   11 v1
2   1   65 v2
1   2   22 v3
1   3   17 v4
2   3   65 v5
1   4   52 v6
1   5   32 v7
1   6   46 v8
1   7   38 v9
1   9   71 v10

(Step 2): Form a linear Diophantine equation with N variables, where N is the number of Sk(na) values found in Step 1. The coefficients of the variables are the Sk(na) values, and the value of the linear expression is to be set equal to n. For example (see table above, where we have named the variables v1 through c10), for n=74 the Diophantine equation is

11v1 + 65v2 + 22v3 + 17v4 + 65v5 + 52v6 + 32v7 + 46v8 + 38v9 + 71v10 = 74.

(Step 3): Solve the Diophantine equation. If the number of variables is relatively small (as it is here) you might wish to find all solutions; otherwise, for the purpose of showing that all integers are interesting, it usually suffices to find ten or twenty solutions. For our example n=74 there are only three solutions, which are:

v3 = 1, v6 = 1   which correspond to 22 + 52 = 74
v1 = 2, v6 = 1     211 + 52 = 74
v1 = 1, v4 = 1, v8 = 1   11 + 17 + 46 = 74

(All vi not shown are equal to zero.)

Each solution to the Diophantine equation describes a linear combination of the Sk(na) values that adds up to n, which is equivalent to having determined that n is an expository number of that type (with the k,a values being those determined in Step 1, and the c values being the vi found in Step 3). For n=74, we now know that it is indeed an expository number, of the following three types:

1(2) + 1(4)
21(1) + 1(4)
1(1) + 1(3) + 1(6)

Step 1 of this procedure conceals a deep, unsolved problem: how large must a be in order to be sure we have found all S1(na) that are < n? This question can be answered if and only if the following conjecture is true:

Digit-Sum Conjecture: For any integer n>1 and any lower bound m>1, there exists an integer A such that S1(na) > m if a > A.

Roughly, this conjecture says that the digit sums of the powers of n grow without bound (but not monotonically - S1(na) as a function of a generally increases, but somewhat erratically). A general proof of this conjecture does not seem to exist in the literature, though it can be proved [4] for certain values of n (namely, those equal to 2, 4, 5, 6, or 8 mod 10).

The lack of a proof of this conjecture means that we do not know we have found all Sk(na) < n in step 1, and so we may not have found all possible expressions of n as an expository number. However, we are still able to complete our construction that shows all integers are interesting, as follows:

(Step 4): Use Theorem 1 to calculate the search limit L for each type found in step 3, and find all expository numbers for each of the types. (There is no uncertainty here, thanks to Theorem 1.) To continue our example, for n=74 we find

Type Solutions
A: 1(2) + 1(4) 38 56 63 74
B: 21(1) + 1(4) 35 41 53 74
C: 1(1) + 1(3) + 1(6) 20 30 63 71 74 81 84 86 92 99

We now notice the following: if we take the intersection of solution sets A and B (or B and C, but not A and C), only n=74 remains. We have thus found a unique property of the integer 74:

74 is the only positive integer equal to
the sum of the digits in its square plus the sum of the digits in its 4th power
and also equal to
twice the sum of its digits plus the sum of the digits in its 4th power.

Sometimes (fairly often, it turns out) it may happen that one of the expository types for n has only one representative (namely, n). In this case, we do not have to say "property P and Q" in our statement, but rather can just say "property P". For example:

135 is the only positive integer equal to
the sum of the digits in its 4th power
plus
the sum of the digits in its 11th power.

To gauge the efficacy of this scheme, we performed Steps 1-4 on all integers from 10 to 1000. Note that there are three possible outcomes for a given n:

  1. We find no solutions to the Diophantine equation, which means (assuming the digit-sum conjecture) that n is not an expository number.
  2. n is an expository number (perhaps of more than one type), but it is not possible to find a set or intersection of sets in Step 4 such that n is the only member.
  3. Steps 1-4 are all successful, yielding a statement of the form "n is the only positive integer of the following type".

Based on our examination of 10 through 1000, we make the following conjectures:

Conjecture 1: The only positive integers of type (i) (i.e, the non-expository numbers) are

11, 13, 14, 16, 19, 29, 37, 44, 55, 67, 73.

Conjecture 2: The only positive integers of type (ii) are:

2, 3, 4, 5, 6, 7, 8, 9, 15, 17, 22, 23, 25, 26, 31, 38, 42, 47,
56, 59, 64, 76, 77, 79, 82, 86, 88, 89, 91, 92, 93, 109

If true, these two conjectures imply our

Main Theorem: All positive integers are interesting.

Proof: The integer 1 is clearly interesting. Interesting properties of the two small lists of numbers in Conjectures 1 and 2 can be easily found (exercise for the reader!). If conjectures 1 and 2 are true, then for any other n, we can construct a statement of the form "n is the only positive integer of expository type T", or "n is the only positive integer of expository type T and also type S". Thus, n is interesting, being the unique number possessing a certain property.

(We should mention a small detail which has thus far been omitted for simplicity: If and only if all the expository types in the uniqueness statement have t=1 (that is, only one term in the sum) and c=1, then the statement must read "n is the only integer >1 with this property". This is because 1 is an expository number of type k(a), for all k and a.)

Table 1 (at the end of this paper) shows an interesting property of every positive integer from 10 to 100, except for those which are not expository numbers of any type (marked in the table with an 'o'). For those integers listed in conjecture 2 (which are expository, but not in a unique way), we have appended an extra condition in order to make the property unique. For example, 64 is the only square greater than one equal to the sum of the digits in its 6th power.

Table 2 is an extension of Table 1 which gives the details for all integers from 10 to 999.

Empirical evidence that conjectures 1 and 2 are true is strong. We examined the first ten sets of one hundred consecutive integers, and in each century found the integer with the smallest number of different representations as an expository number. These values are:

Range Smallest No. of
Representations
1-99 0
100-199 2
200-299 34
300-399 254
400-499 1041
500-599 4340
600-699 11615
700-799 24508
800-899 62264
900-999 145007

These results suggest that every integer > 999 is an expository number in hundreds of thousands of ways, or more. Surely it will be possible to find one or two of these whose intersection of solutions sets is just the value n. Note that we have deliberately limited ourselves to the intersection of two sets. We could take three or more, if necessary, in order to obtain a property that n uniquely satisfies, but we conjecture that this will never be needed.

Many open questions remain for further research, such as obtaining a proof of our primary conjecture, that every integer >109 can be uniquely identified as being of one expository type or the intersection of two types.

That we have used a property (sum of powers of digits of powers of n) which Hardy denounces as uninteresting, in order to show that all numbers are interesting, is a curious fact whose significance we leave for the philosophically-inclined to ponder.

 

References

[1] G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1969, p. 105.

[2] M. Keith, "Power-Sum Numbers", J. Recreational Math. Vol. 18, No. 4 (1985-86), 275-278.

[3] B. L. Schwartz, "Self-Generating Integers", Math. Mag. Vol. 46 (1973), 158-160.

[4] Kurt Foster, private correspondence.


Table 1
Interesting properties of the integers 10 to 100

 10:    10*2(1)
 11: o
 12:    1(2) + 1(1) and 3(1) + 1(1)
 13: o
 14: o
 15: -  1(2) + 1(1) and pq (p,q primes)
 16: o
 17: -  1(3) and prime
 18:    2*1(1)
 19: o
 20:    1(11) + 1(1) + 2(1)
 21:    1(3) + 1(1) and 1(5) + 1(1)
 22: -  1(4) and 11m
 23: -  2(1) + 2*1(1) and prime
 24:    1(3) + 1(1) and 1(2) + 1(1)
 25: -  1(4) and 5m
 26: -  1(3) and pq (p,q primes)
 27:    3*1(1)
 28: -  1(5) and ends in 8
 29: o
 30:    1(17) + 1(1) and 1(10) + 1(1)
 31: -  1(7) and prime
 32:    2(1) + 2*1(2) + 1(1) and 1(2) + 5*1(1)
 33:    1(3) + 1(1) and 1(4) + 1(1)
 34: -  1(7) and 3*1(1) + 1(2) and composite
 35:    1(5) and 1(4) + 2*1(1)
 36:    2*1(1) + 1(2)
 37: o
 38: -  1(4) + 1(2) and ends in 8
 39:    1(3) + 1(1) and 1(4) + 1(2) + 1(1)
 40:    1(13) and 1(11) + 2*1(1) + 1(5)
 41:    2*1(1) + 1(4) and 1(3) + 3*1(1)
 42: -  1(1) + 2*1(2) and 6m
 43:    1(2) + 3*1(1) and prime
 44: o
 45:    1(2) + 1(4) + 1(3)
 46:    1(5) and 1(8)
 47: -  2*1(3) + 1(2) and prime and ends in 7
 48:    1(6) + 1(1) and 1(2) + 1(1) + 1(4)
 49:    4*1(2) + 2(2)
 50:    1(4) + 1(2) + 2(3)
 51:    1(3) + 1(5) + 1(1) and 1(5) + 2*1(2) + 1(1)
 52:    1(4) + 2*1(1) + 1(2) and 2*1(3) + 2*1(1)
 53:    1(7) and 1(6) + 2*1(1)
 54:    2*1(3)
 55: o
 56: -  1(4) + 1(2) and 8m
 57:    1(4) + 1(3) + 1(1) and 1(3) + 1(2) + 1(1)
 58:    1(7) and 1(3) + 3*1(1)
 59: -  1(4) + 1(6) and prime
 60:    1(2) + 2(2) + 1(1)
 61:    1(5) + 3*1(1) and 5*1(1) + 2*1(2)
 62:    1(6) + 2*1(1) and 1(5) + 1(2) + 1(1)
 63:    1(3) + 2(1)
 64: -  1(6) and square
 65:    3*1(1) + 1(5) and 3*1(2) + 1(3)
 66:    1(3) + 1(1) + 1(2) and 1(5) + 1(1) + 1(2)
 67: o
 68:    1(7) and 4*1(3)
 69:    1(5) + 1(1) + 1(2) and 2*1(3) + 1(1)
 70:    2(1) + 3*1(1)
 71:    1(9) and 1(1) + 1(2) + 1(7)
 72:    1(6) + 1(3)
 73: o
 74:    1(2) + 1(4) and 2*1(1) + 1(4)
 75:    1(6) + 1(1) and 1(2) + 1(1) + 1(8)
 76: -  3*1(1) + 1(3) and 19m
 77: -  2*1(3) + 1(2) and 11m
 78:    1(1) + 1(2) + 1(6) and 4*1(1) + 1(2)
 79: -  2*1(1) + 1(4) + 1(2) and ends in 9
 80:    1(3) + 1(1) + 2(1)
 81:    1(1) + 1(7)
 82: -  1(10) and ends in 2
 83:    1(4) + 1(8) and 1(1) + 1(2) + 1(5)
 84:    1(4) + 1(3) + 1(1) and 2*1(5) + 1(1)
 85:    1(10) and 1(2) + 1(5) + 1(1) + 1(4)
 86: -  1(1) + 1(3) + 1(6) and ends in 6
 87:    1(1) + 1(9) and 1(1) + 1(5) + 1(2)
 88: -  4*1(2) and ends in 8
 89: -  2*1(2) + 3*1(1) and prime
 90:    2(1) + 1(1)
 91: -  1(14) and ends in 1
 92: -  1(6) + 1(1) + 1(3) and ends in 2
 93: -  1(9) + 1(1) and ends in 3
 94:    1(10) and 2*1(1) + 2*1(5)
 95:    1(4) + 4*1(2)
 96:    1(6) + 1(1) and 1(2) + 1(1) + 1(7)
 97:    1(10) and 1(5) + 3*1(1)
 98:    1(11) and 1(6) + 2*1(1)
 99:    1(5) + 1(6)
100:    100*1(1) and 99*1(1) + 1(99)