Category: mathematics

Can three cubed numbers be added up to give 42? Until now, we didn’t know

It might not tell us the meaning of life, the universe, and everything, but mathematicians have cracked an elusive problem involving the number 42.

Since the 1950s, mathematicians have been puzzling over whether any integer – or whole number – can be represented as the sum of three cubed numbers.

Put another way: are there integers k, x, y and z such that k = x³ + y³ + z³ for each possible value of k?

Andrew Booker at Bristol University, UK, and Andrew Sutherland at the Massachusetts Institute of Technology, US, have solved the problem for the number 42, the only number less than 100 for which a solution had not been found.

Some numbers have simple solutions. The number 3, for example, can be expressed as 1³ + 1³ + 1³ and 4³ + 4³ + (-5) ³ . But solving the problem for other numbers requires vast strings of digits and, correspondingly, computing power.

The solution for 42, which Booker and Sutherland found using an algorithm, is:

42 = (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³

They worked with software firm Charity Engine to run the program across more than 400,000 volunteers’ idle computers, using computing power that would otherwise be wasted. The amount of processing time is equivalent to a single computer processor running continuously for more than 50 years, says Sutherland.

Earlier this year, Booker found a sum of cubes for the number 33, which was previously the lowest unsolved example.

We know for certain that some numbers, such as 4, 5 and 13, can’t be expressed as the sum of three cubes.

The problem is still unsolved for 10 numbers less than 1000, the smallest of which is 114.

The team will next search for another solution to the number 3.

“It’s possible we’ll find it in the next few months; it’s possible it won’t be for another hundred years,” says Booker.

People interested in aiding the search can can volunteer computing power through Charity Engine, says Sutherland.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Donna Lu*

Calculating Ramsey numbers is so difficult that one mathematician once said he’d rather fight off an alien invasion. Now, mathematicians have made the first major advance in nearly a century.

The key to a successful party is a good mix of people

Mathematicians have made a breakthrough on an incredibly difficult problem in combinatorics, the study of combinations. It is the first significant advance in nearly a century for our understanding of Ramsey numbers, which can be used to describe the minimum size of a party where cliques of a certain size can or can’t exist.

Named after mathematician Frank Ramsey, these numbers deal with the possible relationships between nodes on a mathematical network called a graph, which are a major object of study in maths. The numbers, first discovered in 1928, can be described in terms of people at a party.

Suppose you are hosting a party and want to make sure there is a good balance between people who know each other and those who don’t. Too many mutual acquaintances will form a clique, excluding the others at the party, while too few risks an anti-clique, plagued by awkward small talk.

The question, then, is whether there is a minimum party size that will always produce a clique or anti-clique of a certain size. This is the Ramsey number. For instance, the Ramsey number for a clique of three is six, because in a six-person party you can always find either a clique of three people who know each other, or who don’t.

While this is simple for small numbers, like three, it rapidly becomes very difficult to calculate for larger numbers as the possible combinations grow. It has long been a goal for mathematicians to find a general solution for any Ramsey number.

Paul Erdős, who in 1935 found that the upper limit for any Ramsey number is 4 to the power of the clique size, once said that finding the exact Ramsey number for a clique of six was so difficult that if aliens asked us to try to calculate it, or be destroyed, we would have no choice but to try to attack them first. Since then, generations of mathematicians have tried to significantly improve upon Erdős’s limit, but without much success.

Now, Julian Sahasrabudhe at the University of Cambridge and his colleagues have shown the upper limit for Ramsey numbers can be reduced from 4 to the power of the clique size to 3.993 to the power of the clique size.

“From the outside, it seems sort of silly — like, OK, mathematicians are just trying to improve some number and who cares?” says Sahasrabudhe. “But what’s interesting about these quantitative changes, like the difference between the Erdős bound and previous bounds and our work, is there’s really some quantitative improvement that reflects a deepening of our understanding of the [mathematical] structures.”

Sahasrabudhe and his colleagues first started working on the problem in 2018, at the National Institute for Pure and Applied Mathematics in Rio de Janeiro, Brazil. “It’s a famous problem and we wanted to be ambitious and had some ideas — and it seemed like it was actually going pretty well, which was exciting,” says Sahasrabudhe.

In the first few months the team had sketched out a rough plan of what the proof could look like, he says, but quickly ran into an obstacle, which involved calculating how points clumped together on high-dimensional spheres.

After trying to solve this problem for years and failing, Sahasrabudhe and his team realised they could reformulate their proof to avoid the sphere question entirely, after which things rapidly progressed. “Instead of going over the mountain, we realised we could actually somehow sneak around it,” he says.

The final proof, which is more than 50 pages long and yet to be formally verified, uses elements of Erdős’s original proof, which involved zooming into particular people in arbitrarily large parties and working out the probabilities of their relationships to their neighbours, and also identifying adjacent structures to cliques called “books”, which are easier to find but guarantee that cliques exist.

“For at least the last 50 years, every eminent person in my field has tried quite hard to improve these bounds and failed,” says David Conlon at the California Institute of Technology. “The fact that they have now improved this result is a big deal.”

Now that the barrier of 4 has been broken, other researchers can now use and optimise Sahasrabudhe and his team’s methods to get the number slightly lower than 3.993. “This is a fiendishly hard problem,” says Peter Cameron at the University of St Andrews, UK. “A tiny little improvement like this represents a huge breakthrough in techniques for attacking it.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Alex Wilkins*

See a vertical stripe, what you are seeing is a run of repeated digits, like 7777, 00000, or 222.

One occurs in the top left:

Sequences like this used to be important examples in motivating discussions about the law of the excluded middle. Statements like: ‘the decimal expansion of pi contains the sequence 7777’, according to the law of the excluded middle, are either true or false. But when you are discussing something that has never been calculated, and that might never be calculated, suggests (to constructivists), that the statement may be neither true or false, and that the sequence 7777 may be somewhat like Schrodinger’s cat, neither existing nor not existing.

Now that the digits of pi have been calculated to astounding lengths, and that mathematics exists to uncover the properties of the pi digit sequence, the expansion of pi is no longer an ideal candidate for constructivist thought experiments. Even some exotic questions about the decimal expansion of pi, like the convergence of the brouwer-heyting sequence, have been resolved. Questions about the expansion of pi that illustrate constructivist problems can still be formulated, but they need to be more complicated than they used to be.

Still, I decided to look more closely at this little sequence that occurs (relatively) early on in the pi expansion. The stripe in the pi colour map mentioned above corresponds to the sequence 999999 beginning at digit 763 and ending at digit 768. This was observed in a set of the first 2281 digits of pi.

I decided to push the software a bit further, and load 10000 digits of pi into the document.  (A file with the first 10000 digits of pi formatted for Fathom or TinkerPlots is here.) The plot below shows the digits with 9 in black and all other digits blanked out – dots are single occurrences of a 9, while vertical stripes are runs of repeated 9s.

Using Tinkerplots’ runLength function we can see how frequent runs of various lengths are, and easily identify the longer ones.

Even with 10000 digits, the 999999 sequence at digit 763 was still the longest sequence of repeated digits. The other runner-up sequences in this first 10000 digits are, two instances of 2222, a single run of 8888, and four distinct occurrences of 7777.

The plot at the bottom of this post shows the distribution of even and odd digits in the decimal expansion of pi’s first 10000 digits, and was inspired by a similar plot from Wolfram Math World.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*