Numberphile
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8:29 · 2M views · Nov 6, 2015
Numberphile with Tim Browning on the sum-of-three-cubes problem and why 33 was the lowest unsolved case. The description notes a later solution for 33.
Linked in threads on sums of cubes and Diophantine equations as an accessible introduction.
3 comments from Hacker News
It's an open question that has been studied for thousands of years. This type of equation (a polynomial with integer coefficients) is called a Diophantine equation; you can look it up on Wikipedia. Or watch videos about it: this video
I only recently heard what I think is a pretty solid answer to this question [1] that put it this way: We've pretty much got addition nailed down. We're pretty solid on multiplication. However, we are surprisingly weak mathematically when we try to put the two together; you can write down some extremely small math statements that baffled mathematicians for centuries, and for such statements that we have solutions for, they often get into very, very complicated math. So you have things like Fermat's Last Theorum, or if you prefer, the Question: For a^n + b^n = c^n, are there any solutions with n > 2 for integers? Note how it's just multiplication and addition on the integers. There are simpler questions, but they don't get much simpler. There's no real analysis, transcendental numbers, imaginaries or higher-dimensional numbers, infinities, and so on, at least not in the question itself; it's literally so simple you can reasonably explain it to a middle schooler. It took mathematicians centuries, and the simplest version of the current proof is still PhD-level work to understand. (I suspect even a Master's student in math would have to carefully craft their entire post-doc education for the purpose of understanding this exact proof to be able to say they fully understood it, with no lemmas taken on faith, and I'm still not sure they could make it without a lot of independent study.) You have things like "Can all of the numbers of a certain form be created via 'a^3 + b^3 + c^3'?
Numberphile video on the topic with Andrew Booker: [LINK] which is a followup from the 2015 episode this video when the 33 problem was unsolved. The announcement elicited this Mathoverflow discussion about "mic drop" moments in mathematics: [LINK]