The Strong Law of Small Numbers

The strong law of small numbers was a scientific article originally written by Richard K. Guy in 1988. In the field of mathematics, there is a lot of attention around exotic numbers and large numbers such as infinity and googol. However, Guy was more concerned about the problems that revolve around smaller numbers. Small numbers, in this context, refer to the numbers that are in the relatively countable range.

He proposed (with a sense of humour) that there are not enough small numbers to meet the many demands made of them. He went on to intimidatingly (again, with a sense of humour) state the following:

1. “Capricious coincidences cause careless conjectures.

2. Initial irregularities inhibit incisive intuition.”

— R.K. Guy

It is not a problem if you have difficulty understanding these statements. It will become clearer as you read along. Guy essentially wanted to show that patterns are misleading when we are dealing with small numbers. He went on to cite a few examples where small numbers seem to exhibit patterns that might not be true after all.

In this article, I try to explore and demonstrate some of these examples so that we may learn from Guy’s genius. In doing so, we would gain more appreciation for the fundamental challenges we face with day-to-day numbers. It is often the case that these challenges are directly transferable to everyday life situations. We often mistake the frequency of occurrence of an event for absolute truth.

This essay is supported by Generatebg

Perfect Squares and Misleading Facts

Let’s start by considering small (relatively countable) numbers that are perfect squares. 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are perfect squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 respectively.

What this goes to show is that ten per cent of the first hundred numbers are perfect squares. Based on this frequency, it is easy for us to assume that perfectly square numbers constitute roughly ten per cent of all numbers. Let’s test that assumption.

Let us consider the first thousand numbers. It turns out that only around 3% of the first thousand numbers are perfect squares. If we keep increasing the number range, the percentage only reduces.

In fact, as the number range approaches infinity, the percentage of perfect squares approaches zero. Anyone living in Infinity-Land might think that perfect squares do not exist. This is the inverse fallacy to what we started with — that perfect squares comprise around ten per cent of all numbers.

Fermat’s Prime Patterns

Consider the following question and pattern series:

Here, we are considering positive values of integers for n. We see that we have obtained prime numbers as results so far. But as we continue, we get the following result:

The result turns out to be not a prime number. Furthermore, we do not know if this set is finite. These numbers are known as Fermat primes, named after the person who made this claim, Pierre de Fermat. The error in the claim was first proved by Leonhard Euler.

More Prime Patterns

Let us consider the following claim and the pattern that ensues:

Is 3…31 always prime?

31 — Prime Number.

331 — Prime Number.

3331 — Prime Number.

33331 — Prime Number.

333331 — Prime Number.

So far, it seems pretty convincing that this series leads to prime numbers. Let us keep going and see if we come across an exception.

3333331 — Prime Number.

33333331 — Prime Number.

333333331 = 17*19607843 — Not Prime!

You see, this is how deceiving it can get. This just goes to show that lack of evidence does not lead to truth.

Lack of evidence of an exception and evidence of lack of an exception are two very different things.

Let’s cover one more example.

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Deleting Numbers and Taking Partial Sums

Consider the following series, where you delete every other number (even numbers in this case), and then take partial sums.

It appears that this series leads to perfect squares. The question now is if this is true. Contrary to what we have seen so far, it turns out that this series is, in fact, true!

This process of analysis is known as Mössner’s theorem, named after Alfred Mössner. Initially, it was no theorem, but just a claim. Later on, Oskar Perron proved it (for details, please see references at the end of the article), and the concept was further generalized by ensuing mathematicians.

What is The Strong Law of Small Numbers Trying to Say?

Guy had a terrific sense of humour. What he tried to convey with his little article was more of a philosophical and statistical point. It can be summarized in his own words:

“You can’t tell by looking.”

— R.K. Guy

Don’t worry; I’ll elaborate. Guy suggests any person working with small numbers to be very suspicious about patterns observed. If you observe a pattern and the sample size is small, do not jump to conclusions.

The pattern that occurs in a small sample may appear tempting, but it may or may not be true. There is no way of proving such a hypothesis just by looking at the pattern. This is essentially what Guy means by the sentence that says that you can’t tell by looking. This so-called strong law is arguably the inverse of the statistical law of large numbers.

When I phrase it this way, it makes sense straight away. I wouldn’t blame you if you think that this is a simple point to remember. But the genius of Guy was to wrap his statement into powerful counter-intuitive examples. Just when we think that we know something, we become most susceptible to mistakes. There are a lot more of these profound mathematical examples in his original scientific article, which I’ll link in the references below.

If you still think that this should not be as much of a big deal as it is being made out to be, just know that even top professionals make mistakes with small numbers. The small numbers are a tricky bunch. You would be wise to stay alert and prove your hypotheses before jumping to conclusions!

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References: Richard K. Guy (scientific article), Alfred Mössner (scientific article — language: German), and Oskar Perron (scientific article — language: German).

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Further reading that might interest you: What Really Happens When You Measure A Coastline? and Are We Living In A Simulation?