Find the average number of ways we can express a natural number 𝑛 as the sum of two squares

Number Theory Level 3

The average number of ways we can express a natural number 𝑛 as the sum of two squares…

We are not looking for an answer for some particular 𝑛. We are not looking for a general formula.

We just want to know… on an average what is this equal to.

We’re going to be fairly generous about what we count here,

So for 𝑛=1 -> there are four ways:

0^2+1^2,

0^2+(−1)^2,

1^2+0^2

and (−1)^2+0^2.

There are another four solutions for 𝑛=2:

But then there are no ways to express 3 as the sum of the squares of two natural numbers.

In fact (I claim, with exactly zero formal proof) most numbers can’t be expressed like this — but then there are 160 ways to express 4,005,625 as the sum of the squares of two natural numbers.

You will find however that there are 3,184,193 un-expressable numbers along the way.

So what does that mess of zeroes and hundreds average out at ?

The answer is 3.14159.

1 solution

Mark Hennings
Mar 20, 2019

It is a property of the sum of squares function $r_2(n)$ that $\sum_{k=1}^n r_2(k) \; = \; \pi n + O(\sqrt{n}) \hspace{2cm} n \to \infty$ and hence $\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^n r_2(k) \; = \; \boxed{\pi}$ which is the nearest thing we can get to calculating the expected value of $r_2$ .

Find the average number of ways we can express a natural number 𝑛 as the sum of two squares

The answer is 3.14159.

1 solution

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