One of the Tiniest Proofs Ever Published
Leonhard Euler was the Mozart of mathematics. If you’ve ever seen the number e, you know what the “e” stands for? Euler. You know the “e” in e-mail? That doesn’t stand for Euler, but it should. Look at this guy:
Did you feel that? If not, look again. Do you feel it now? That’s Euler’s math prowess radiating from the painting, through your screen, and straight into your bones.
On top of being a brilliant mathematician, Euler was insanely prolific. Euler wrote so much that the still-incomplete (!) collection of his works has grown to 78 volumes. He was like Isaac Asimov or Pablo Picasso in terms of sheer output, and he made vital contributions to graph theory, calculus, analysis, topology, even music theory (and notation: today’s use of π, f(x), e, i, and Σ are all attributed to Euler). He’s considered one of the greatest mathematicians in all of history.
With that out of the way, let’s talk about that time Euler was wrong.
Euler’s Sum of Powers Conjecture, Explained
Among other things, Euler was interested in powers of numbers, and in 1769 he made a conjecture now called Euler’s sum of powers conjecture (or just Euler’s Conjecture). In general, a conjecture is a math statement that has not been proven yet, and a theorem is a big result that’s been proven. So Euler was making a sort of educated guess.
To understand Euler’s conjecture, let’s start by looking at the Pythagorean theorem:
a²+b² = c²
There are lots of numbers that satisfy this equation, like (3,4,5) or (6,8,10). There are also numbers that satisfy the similar equation below:
a³+b³+c³ = d³
For example, (3,4,5,6). However, this equation:
Has no solutions¹. So now let’s get general. Suppose you have a number b raised to the 4th power (b⁴). You want to add some other numbers, also raised to the 4th power, so that they sum up to b⁴:
a₁⁴ + a₂⁴ + a₃⁴ + a₄⁴ … aₙ⁴= b⁴
How many different a numbers would you need? Euler guessed that this equation is impossible without at least 4 a’s (n=4). If instead we raised b and all the a’s to the 6th power, Euler would tell us that we need at least 6 a’s for the equation to be possible (n=6). And in general, Euler’s Conjecture states that if b and all the a’s are raised to the nth power for some number n, then we need at least n different a’s to make the equation possible. Or to write it in math notation:
a₁ᵏ+a₂ᵏ+…+aₙᵏ = bᵏ implies n≥k.
That is Euler’s sum of powers conjecture.
Disproving Euler’s Conjecture with a Supercomputer
Euler’s conjecture was disproven in just 2 sentences, almost 200 years later.
One of the biggest computer companies in the 1960s was the Control Data Corporation, which in hindsight is a terrifying name for a company. They were one of the “biggest” as in company size but also “biggest” as in they made huge supercomputers like the CDC 6600. The CDC 6600 was the fastest supercomputer from 1964 to 1969, and it could compute 3 million FLOPS (floating point operations) per second. For reference, a Nintendo DS can compute 600 million FLOPS per second, compared to the CDC 6600’s 3 million. Oh also the CDC 6600 cost $20,244,689.49 (adjusted for inflation in 2021).
But the CDC 6600 was revolutionary for its time, especially because you could use it as a kind of giant calculator. So in 1966, that’s what mathematicians Leon J. Lander and Thomas Parkin did. They took this equation…
a₁⁵+a₂⁵+a₃⁵+a₄⁵ = b⁵
…and they searched for solutions with the CDC 6600. Euler’s conjecture says this equation should have no solutions, since each number is raised to the 5th power and there are only 4 a’s. It should be impossible. But rather than getting a big computerized image of Euler wagging his finger at Lander and Parkin², the two mathematicians found this:
27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵
They found a solution! And that’s enough to disprove Euler’s conjecture. Lander and Parkin published their findings in this two-sentence-long paper. You could paraphrase the paper like this:
- Sentence 1: Hey guess what, 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵
- Sentence 2: This is a counter-example to Euler’s conjecture ok bye³.
The Proof by Counterexample
If a movie was two minutes long, you would call it a ripoff. If a book was two sentences long, you would call that a scam. Can a math proof this short still be legitimate? Yes. Lander and Parkin’s paper is a great example of a proof-by-counterexample.
Proof-by-counterexample is one of many ways to structure a math proof. This proof is only used to disprove a statement (it may make more sense to call it a disproof by counterexample), and it’s often used on statements that assert something is “always true” or “never true,” like “no pink dogs exist” (versus something less-extreme like “a few pink dogs exist”).
To disprove a statement by counterexample, all you have to do is find a clear example where the statement is wrong. If I tell you “no black swans exist,” then you find a black swan, that’s a counterexample. If I tell you “all people in the conference room are named Dave” and you find one person named Ophelia, she is a counterexample (and a great project manager). You do not need to check the name of every person in the room anymore, because you only need to find one counterexample to prove me wrong.
This is why Lander and Parkin’s proof was so short: since Euler claimed that there were no solutions to a⁵+a⁵+a⁵+a⁵ = b⁵, they only needed one solution as a counterexample. Since then, more counterexamples have been found, like this one found by Noam Elkies in 1988:
2,682,440⁴ + 15,365,639⁴ + 18,796,760⁴ = 20,615,673⁴
Leonhard Euler was a titan of mathematics, one of the greatest, but he was still human. Even he cannot evade the occasional mistake. Not long ago, I wrote that I couldn’t think of a famous example where mathematicians thought one thing was true but then the opposite was true instead. Professor Howard Sporn very generously sent me Lander and Parkin’s paper as, well, a counter-example. In just two sentences, Lander and Parkin not only show us how a computer search disproved a 200-year-old statement, it shows us one of math’s many graceful proof structures: the counter-example.
¹: But this wasn’t fully proven until 1995, long after Euler’s death. It’s a long story.
²: I have not been able to confirm nor deny that this was not a feature in the CDC 6600.
³: My goal is to rewrite this paper as a haiku. If you can rewrite the paper as a haiku, please send it to me.