Hardy's Example of Non-Serious Theorems

There are two numbers below $10000$ that are multiples of their reversals:

$8172 = 4\times 2718$ and $9801=9\times 1089.$

According to Joe Roberts, this fact has acquired a certain kind of notoriety from Hardy's statement in his Mathematician's Apology:

These 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 now interesting - merely tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and proofs, which are not capable of any significant generalization.

Joe Roberts quotes from an article by M. Beech:

... my Apple IIc confirmed that for all integers up to 10000 the only which are multiples of their reversals are $8712$ and $9801$ ... a few more hours of running time were required to test all integers up to 100000 ... Between 10000 and 100000 the only integers which are integral multiples of their reversals are $87912$ and $98901$ with $87912=4\times 21978$ and $98901 = 9\times 10989.$

The obvious generalization, Roberts adds, holds where the single central $9$ is replaced by a string of $9\text{'s}.$ Thus the "computer conjecture" is that this exhausts all possible such numbers.

On November 13, 2012, Gareth Williams presented a seminar "Reverse Divisors: 1089 and all that follow" at the Open University, and his article (with Roger Webster) under the same title appeared next year in Mathematical Spectrum. (If $9801$ is a reverse multiple, $1089$ is a reverse divisor.) Without announcing the result right away they invited the readers to try finding the first reverse divisor, with a calculator if necessary. They then added,

For those of you who were successful, and those who were not, please read on and join us on a mathematical journey, from knowing nothing about reverse divisors to knowing everything! We found it exciting, we hope that you will do so too.

In the paper that appeared in 2013 they proved that the "computer conjecture" was conservative in that they found additional reverse divisors; they also observed a few additional features of the reversibles. They stated their theorem thus:

Theorem 1.1

The quotient of a reverse divisor is either $4$ or $9.$ The two $n$-digit $(n\ge 4)$ numbers

$11\times (10^{n-2}-1)=1099\ldots 9989$ and $22\times (10^{n-2}-1)=2199\ldots 9978,$

each with $n-4$ central digits $9$, are reverse divisors with respective quotients $9$ and $4.$ No reverse divisor has fewer than four digits.

I side with Hardy as regards the proof. It's not long - just half a page - but not illuminating. However, the coming next corollary is a curiosity

Corollary 1.2

The product of a reverse divisor and its reverse is a square.

In Section 2 of the article they deal with the divisors whose quotient is $9.$ The expand the set of the divisors:

$1089,$ $10989,$ $109989,$ $1099989,$ $10891089,$ $10999989,$ $108901089,$ $109999989,$ $1089001089,$ $1098910989,$ $1099999989,$ $10890001089,$ $1098010989,$ $10999999989,$ $108900001989,$ $108910891089,$ $10989001089,$ $109989109989,$ $109999999989\ldots$

Analogous results are valid for the quotient of $4:$

$2178,$ $21978,$ $219978,$ $2199978,$ $21782178,$ $21999978,$ $217802178,$ $219999978,$ $2178002178,$ $2197821978,$ $2199999978,$ $21780002178,$ $21978021978,$ $21999999978,$ $217800002178,$ $217821782178,$ $219780021978,$ $219978219978,$ $219999999978\ldots$

These have been placed in a separate paper.

Following David Huitson, they also observe that some other fraction in which the denominator is the reverse of the numerator have the same curious and interesting cancellation property:

$\displaystyle\frac{2}{3}=\frac{4356}{6534}=\frac{43956}{65934}=\frac{439956}{659934}=\ldots$

and

$\displaystyle\frac{3}{8}=\frac{27}{72}=\frac{297}{792}=\frac{2997}{7992}=\ldots$

Towards the end, they mention that the students seem to be much taken with reverse divisors and also look into the situation in bases other than decimal.

References

1. M. Beech, A computer conjecture of a non-serious theorem, Math. Gazette, 74 (1990) 50-51
2. G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1994, 104-105
3. J. Roberts, Lure of the Integers, MAA, 1992, 36-37
4. R. Webster, G. Williams, On the Trail of Reverse Divisors: 1089 and All that Follow, Math Spectrum 45 (2013) n 3, 96-102