Is every integer the sum of four perfect cubes?

Interestingly, no one knows. It’s conjectured that the answer is yes, but no one yet has been able to prove this.

“Good tests kill flawed theories; we remain alive to guess again.” — Karl Popper

“There are two possible outcomes: if the result confirms the hypothesis, then you’ve made a measurement. If the result is contrary to the hypothesis, then you’ve made a discovery.” — Enrico Fermi

“This particular thesis was addressed to me a quarter of a century ago by John Campbell, who … told me that all theories are proven wrong in time. … My answer to him was, ‘John, when people thought the Earth was flat, they were wrong. When people thought the Earth was [perfectly] spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.’ The basic trouble, you see, is that people think that ‘right’ and ‘wrong’ are absolute; that everything that isn’t perfectly and completely right is totally and equally wrong. However, I don’t think that’s so. It seems to me that right and wrong are fuzzy concepts.” — Isaac Asimov, The Relativity of Wrong, 1988

By Wikimedia user Cmglee, a visual proof that 3³ + 4³ + 5³ = 6³:

This value, 216, is sometimes called Plato’s number because it seems to correspond to this enigmatic passage in the Republic:

Now for divine begettings there is a period comprehended by a perfect number, and for mortal by the first in which augmentations dominating and dominated when they have attained to three distances and four limits of the assimilating and the dissimilating, the waxing and the waning, render all things conversable and commensurable with one another, whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation, the one the product of equal factors taken one hundred times, the other of equal length one way but oblong,-one dimension of a hundred numbers determined by the rational diameters of the pempad lacking one in each case, or of the irrational lacking two; the other dimension of a hundred cubes of the triad. And this entire geometrical number is determinative of this thing, of better and inferior births.

Unfortunately, Plato’s meaning is far from certain. Other values that have been proposed (with various rationales) include 1,728, 3,600, 5,040, 8,128, 17,500, 760,000, and 12,960,000. Cicero politely described the question as “obscure”; it remains open.

In the December 2024 issue of Recreational Mathematics Magazine, Illinois State University mathematician Sunil Chebolu demonstrates that the digits of π can be generated using the positions of stars.

The probability P(x, y) that two randomly chosen positive integers, x and y, are relatively prime is 6/π². The stars in the night sky appear to be distributed randomly on the celestial sphere, and this suggests a novel experiment: Use the pairwise angular distances between stars to simulate a large random sample of numbers, then pick random pairs and determine the proportion of relatively prime pairs. Equating that with the theoretical probability above should generate an approximation of π.

In Nature in 1995, University of Aston mathematician Robert Matthews used the positions of 100 stars to compute π to within 0.5% of its correct value (3.12772). By expanding the sample to 1,000 stars, Chebolu got an average estimate of 3.141540567, an error of less than 0.002%.

“The above method can also be viewed as a statistical hypothesis test,” Chebolu notes. “Since the error in the estimate for π obtained using this method is pretty low, one may also argue that it supports the hypothesis that the bright stars are randomly distributed on the celestial sphere.”

(Sunil K. Chebolu, “Baking Star π,” Recreational Mathematics Magazine 11:19 [December 2024], 67-70.)