For once - four ones!

1111

Not only is 1/1/11 the start of the New Year, but- without the strokes - we have 1111: a palindrome (a number which reads the same if the order of the digits is reversed). It is also a repunit: a whole number none of whose digits is other than 1. The first few are: 1, 11, 111, 1111, 11111 …

I have always thought of such numbers as oners (it helps to name things in the mathematical garden) but since 1966 when Albert H. Beiler coined the term repunit (a portmanteau of REPeated and UNIT) others have used this designation, so must I if I want to communicate and be understood.

There is something aesthetically interesting about numbers exhibiting a high degree of symmetry, so such numbers may seem to some special in non-mathematical ways. To some this manifests itself as a mystical feeling. For example, Uri Geller writes about the significance of the 11.11 phenomenon. Certain numbers seem to crop up significantly more often than others in apparently random contexts.

To many people 11.11 is just a time on the clock, and, worse still, just one among many. We pay more attention to it than to other less striking clock readings- those not having any obvious and immediate symmetrical feel- because it is a special number, not because that time has any special significance (though I do think that since it consists of two elevens it ought to be called ”elevenses”.

I myself, if I ever catch the microwave clock showing 3.14, pause and eat the nearest available approximation to a piece of pie, often saying out loud: ”pi time: it's 'high time for pie' time”. I have regarded this ritual as evidence of playfulness (homo ludens) and an excuse to eat pie rather than anything of cosmic significance. How convenient to be able to blame one's indulgence on some supernatural ukase issued through medium of the microwave. But would anyone believe it?

After all, the appearance of a special number on an object does not make the object special. Nor does it mean that a collection of otherwise arbitrarily-numbered objects that happen to share the same special number are special in their own right.

Besides being patterned, repunits may generate other patterns. For example, 1111 X 1111 = 1234321, a result which some describe as being like a pyramid. This is not, strictly, good as an analogy, as pyramids are 3-D figures. It looks rather like a stepped triangle. In mathematics care is taken to be careful to be accurate with definitions, as a careless use of words and sloppy designations can really lead one up the garden path.

The fact that squaring 1111 produces a stepped triangle is a feature of the squares of all repunits from 1 to 111111111. But thereafter carries become involved in the calculation and the perfection of the pattern is disrupted. (1111111111 X 1111111111 = 1234567900987654321).

Can one use repunits to make patterns that do not break down? Yes. For example, all repunits with an even number of digits may be expressed in the form:

11 = 2 + 3^2
1111 = 22 + 33^2
111111 = 222 + 333^2

and this pattern works not just for the first ten but for arbitrarily long repunits with an even number of digits. This shape is reminiscent of the silhouette of a stepped ziggurat or a sawn of stepped triangle.

Such numerical regularities are interesting and attractive (hence their use in recreational mathematics and puzzles) but their explanation is to be found routinely and quite ”down-to-earthily” in the structural articulation of the multiplication and addition of digits. If we were to watch a mechanical device computing the result of such a calculation we would see the cogs entering regular cycles as a result of the relations between the number of cogs and the digits of the numbers. The eye is good at spotting such patterns, for reasons connected with the evolution of the visual system.

Repunits are of interest not only in recreational maths but also in number theory. In the 19th Century they were studied in relation to repeating decimals, and more recently in relation to the theory of prime numbers.

Points to Ponder

1. An n-digit repunit cannot be prime if n is composite. Why?

2. Show that 1 is the only repunit that is a perfect square.

3. Show that for any odd number you can find a repunit that is a multiple of it.

4. 1/27 = 0.037037037... and 1/37 = 0.027027027... Is this a coincidence?

5. Some mathematicians initially thought that the idea of a repunit was too arbitrary to have any deep mathematical significance as it had too much to do with decimal digits. Were they right?