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Going out on a limb, I’m betting that you don’t know what the Hasse–Minkowski local-global principle for quadratic forms is. Don’t worry. That makes four of us—you three who read this column and me. But even if I don’t know what it is, a recently-published paper tells me that “using this precise description, we see that a density of 0% of integers are the sum of two rational squares".

There’s something essentially mathematical there. You might interpret it to mean there are no such integers—0%, after all. But wait! Take 109, which is 100 + 9, both of which are rational and square (of 10 and 3, respectively). And you can easily find plenty more such pairs. Clearly, 0% is wrong. So what’s this “precise description" about?

Here’s what it means. Sure, there are a lot of integers that can be expressed as the sum of two rational squares: in fact, an infinity of them. But if you take the number of them that’s below a given threshold—a million, say—the fraction these particular integers form of that million is tiny. Push the threshold up, and the fraction gets vanishingly small—effectively, 0%.

To understand this, start by recognizing that this idea of 0% density is hardly an unusual characteristic of numbers. Consider the squares themselves. Up to 100, there are 10 squares or 10%. Push that threshold to 1,000, and we have 31, or 3.1%. Raise to a million? Just 1,000 squares, or 0.1%. That fraction is quickly getting vanishingly small—so indeed, the density of squares is effectively 0%. Since that’s so, numbers that are the sums of pairs of squares also get sparse, if less quickly.

Anyway, there’s an easily understood test, dating from the 1600s, for whether an integer is the sum of two squares. The reality is that the vast majority of integers fail the test. Another reality is that if an integer passes the test, the two squares must themselves be integers (like 100 and 9 above) and not fractions (which are also rational).

But if all that’s interesting, or even if it isn’t, this paper I mentioned is concerned with squares only in passing. It is actually about summing cubes (Integers expressible as the sum of two rational cubes, Levent Alpöge, Manjul Bhargava, and Ari Shnidman, https://bit.ly/3FxYpwq, 19 October 2022). Now, why would a trio of accomplished mathematicians study this? Because, in startling contrast to squares, the fraction of integers that can be expressed as the sum of two rational cubes is far from vanishingly small. It’s significantly more than 0%.

This is so because—again, in contrast to squares—the rational cubes we add together to make integers don’t have to themselves be integers. Pairs of fractions can do the job. One example: 13 = (2/3)3 + (7/3)3. If there weren’t such pairs of fractions, the density of these particular integers—like with sums of squares—would be 0%.

Why the stark difference between squares and cubes? That’s a conundrum by itself. But here’s the kicker: we know that fraction is higher than 0%, but we don’t know exactly what it is. In fact, pinning it down is one of number theory’s more hoary, grey-haired puzzles that mathematicians have been gnawing at for years. We do know that 62 of the integers below 100 can be expressed as the sum of two rational cubes: 62%. As we get to larger numbers, it gets harder to identify the ones with this property and nail down that fraction. So far, mathematicians have only managed informed conjectures.

For example, some mathematicians showed in 2010 that if the well-known Birch-Swinnerton-Dyer conjecture—as with Hasse–Minkowski, never mind what it is—is ever proved, 59% of the integers below 10 million can be expressed as the sum of two rational cubes. Others have theorized that half of all integers have this characteristic. As the paper above puts it: “it is natural to conjecture that the integers that can be expressed as the sum of two rational cubes should have natural density exactly 1/2."

Still, it remains just another conjecture.

Though this trio of mathematicians has now managed to bookend the conjectures, they don’t know what exactly the fraction is. They know it is “strictly positive and strictly less than 1". Or as their theorem states: “A positive proportion of integers are the sum of two rational cubes, and a positive proportion are not." The first of those positive proportions, they proved, is not less than 2/21; the second, not less than 1/6.

There lie the bookends. The fraction of all integers that are the sum of two rational cubes is somewhere between 2/21 and 5/6.

Perhaps you’re indifferent to this. But put yourself in a mathematician’s shoes for a moment, and then give it a thought. This number you’re chasing might have been zero, like with squares. Then you realize it is almost certainly more than zero, though you can’t prove it. It might be 59%, but we need that other conjecture to be resolved, and that’s not happened. Many of your professional colleagues suggest that it’s “natural" to think it’s 50%. But you can’t prove that either. So really, the goal remains elusive.

Enter Alpöge, Bhargava and Shnidman. No, they don’t have a fixed number, either. But they do show that the fraction you’re in search of has to be between 2/21 and 5/6.

It’s as if you’re searching for a book on football in a large library spread across multiple floors in a building shaped like Texas (yes, there is one such). You know it’s in there, but you don’t know where. Someone asks: “Does it have a red cover? For some reason, this library stores all red books on the third floor." But you don’t know if it’s red.

Enter three accomplished football fans back from the World Cup. “We’ve just checked," they say. “All books on football are between the 5th and 7th floors."

I’d call that progress.

So even though I’m no mathematician and I grasp only a small fraction of their paper, I say to Alpöge, Bhargava and Shnidman: for this essentially mathematical musing, thank you!

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun. 