All Numbers are InterestingA 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

"

nis the only positive integer with the propertyP(n)"for every positive integer

n. On the one hand, it is trivial to find a suitable propertyP- 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 propertyPfor which it is at least modestly unobvious that onlynpossesses 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 = 1

^{3}+ 5^{3}+ 3^{3}, 370 = 3^{3}+ 7^{3}+ 0^{3}371 = 3

^{3}+ 7^{3}+ 1^{3}, 407 = 4^{3}+ 0^{3}+ 7^{3}

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 notationS_{k}(n) to stand for the sum of thekth powers of the decimal digits ofn, so this asks for numbers for which

n=S_{k}(n)In [2] this notion is further generalized to a sum of sums of powers of the digits of

n; that is:

tn=Σ

c_{i}S_{ki}(n^{ })i=1We now define a further generalization, in which the sum of the powers of digits of various

powersofnare allowed:

tn=Σ c_{i}S_{ki}(n^{ ai})(1) i=1where the

c_{i},k_{i},a_{i}are all positive integers. If a set of parameters {c_{i},k_{i},a_{i}}can be found such that equation (1) holds, then we callnanexpository 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 =

S_{3}(666^{2}) +S_{1}(666^{3})That is, 666 is equal to the sum of the cubes of the digits in its square (666

^{2}= 443556, and the sum of the cubes of these digits is 4^{3}+ 4^{3}+ 3^{3}+ 5^{3}+ 5^{3}+ 6^{3}= 621) plus the sum of the digits in its cube (666^{3}= 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

S_{2}(8112^{23})We refer to an expository number with a given set of parameters {

c_{i},k_{i},a_{i}}as being oftypec_{1}·k_{1}(a_{1}) +c_{2}·k_{2}(a_{2}) + ... +c_{t}·k_{t}(a_{t}), with ac_{i}being omitted if it equals 1. So 666 is of type 3(2) + 1(3) and 8112 is of type 3·2(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:Letnhaveddecimal digits. We show that ifdis 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 eachn^{ ai}equal 9. Since there are (at most)da_{i}digits inn^{ ai}, the maximum possible value ofS_{ki}(n^{ai}) isda_{i}9^{ki}. On the other hand, the minimum possible value of the LHS is 10^{d-1}. So equation (1) cannot be satisfied if

t10 ^{d-1}>Σ d c_{i}a_{i}9^{ki}=dKi=1or

10

^{d-1}/d>K.Since

Kis a constant and the left-hand side increases monotonically withd, then this inequality holds for alldsufficiently 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), wheref(n) is an arbitrary function of the digits ofn.)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

Dor more digits, and the maximum value that formula (1) can give forD-1 digits is (D-1)K, which we denote byL. Only the integers from 1 toLneed to be checked.Example: For type 3(2) + 1(3) discussed above,

K= 2·9^{3}+3·9^{1}=1485. Thus the inequality holds whenD=5 (since then 2000 > 1485), and the search limit isL= (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

S_{1}(n^{a}) that are<n.Then, for each value ofaused there, computeS_{k}(n^{a}) fork=1, 2, 3, ... untilS_{k}(n^{a}) >n. We now have all possible values ofS_{k}(n^{a}) that are<n. For example, forn=74 there are tenk,acombinations with sum<74:

kaS_{k}(74^{a})1 1 11 v_{1}2 1 65 v_{2}1 2 22 v_{3}1 3 17 v_{4}2 3 65 v_{5}1 4 52 v_{6}1 5 32 v_{7}1 6 46 v_{8}1 7 38 v_{9}1 9 71 v_{10}(Step 2): Form a linear Diophantine equation with

Nvariables, whereNis the number ofS_{k}(n^{a}) values found in Step 1. The coefficients of the variables are theS_{k}(n^{a}) values, and the value of the linear expression is to be set equal ton. For example (see table above, where we have named the variablesv_{1}throughc_{10}), forn=74 the Diophantine equation is11

v_{1}+ 65v_{2}+ 22v_{3}+ 17v_{4}+ 65v_{5}+ 52v_{6}+ 32v_{7}+ 46v_{8}+ 38v_{9}+ 71v_{10}= 74.(Step 3): Solve the Diophantine equation. If the number of variables is relatively small (as it is here) you might wish to find

allsolutions; otherwise, for the purpose of showing that all integers are interesting, it usually suffices to find ten or twenty solutions. For our examplen=74 there are only three solutions, which are:

v_{3}= 1,v_{6}= 1which correspond to 22 + 52 = 74 v_{1}= 2,v_{6}= 12·11 + 52 = 74 v_{1}= 1,v_{4}= 1,v_{8}= 111 + 17 + 46 = 74 (All

v_{i}not shown are equal to zero.)Each solution to the Diophantine equation describes a linear combination of the

S_{k}(n^{a}) values that adds up ton, which is equivalent to having determined thatnis an expository number of that type (with thek,avalues being those determined in Step 1, and thecvalues being thev_{i}found in Step 3). Forn=74, we now know that it is indeed an expository number, of the following three types:

1(2) + 1(4) 2·1(1) + 1(4) 1(1) + 1(3) + 1(6) Step 1 of this procedure conceals a deep, unsolved problem: how large must

abe in order to be sure we have found allS_{1}(n^{a}) that are<n? This question can be answered if and only if the following conjecture is true:

Digit-Sum Conjecture:For any integern>1 and any lower boundm>1, there exists an integerAsuch thatS_{1}(n^{a}) >mifa>A.Roughly, this conjecture says that the digit sums of the powers of

ngrow without bound (but not monotonically -S_{1}(n^{a}) as a function ofagenerally 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 ofn(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

S_{k}(n^{a})<nin step 1, and so we may not have foundallpossible expressions ofnas 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

Lfor each type found in step 3, and findallexpository numbers for each of the types. (There is no uncertainty here, thanks to Theorem 1.) To continue our example, forn=74 we find

TypeSolutionsA:1(2) + 1(4)38 56 63 74 B:2·1(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 alsoequal 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

nhas only one representative (namely,n). In this case, we do not have to say "propertyPandQ" in our statement, but rather can just say "propertyP". 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:

- We find no solutions to the Diophantine equation, which means (assuming the digit-sum conjecture) that
nis not an expository number.

nis 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 thatnis the only member.

- Steps 1-4 are all successful, yielding a statement of the form "
nis 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) are11, 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, 109If 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 othern, we can construct a statement of the form "nis the only positive integer of expository typeT", or "nis the only positive integer of expository typeTand also typeS". Thus,nis 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) andc=1, then the statement must read "nis the only integer >1 with this property". This is because 1 is an expository number of typek(a), for allkanda.)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

smallestnumber of different representations as an expository number. These values are:

RangeSmallest No. of

Representations1-99 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 valuen. 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 thatnuniquely 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 1Interesting 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)