October’s topics:
- Another Mersenne prime
- Concepts vs. practice in Kindergarten math education
- “Why physics is unreasonably good at creating new math”
Another Mersenne prime.
On October 21, the Great Internet Mersenne Prime Search (GIMPS) announced the discovery of a new largest prime number. For number theorists, the set of prime numbers is like the periodic table for chemists: the irreducible elements of which everything else is constituted. We’ve known since Euclid that there are infinitely many of them; the Prime Number Theorem (1896) states that the number of primes below $x$ is asymptotic to $x/\log(x)$, so that among the integers primes are sparser the farther out you go. Mathematicians have taken it as a challenge to find bigger and bigger examples.
The new largest prime is a Mersenne prime, one of a class of prime numbers that can be traced back to Euclid. They come up in his work on perfect numbers. (A perfect number is one that equals the sum of all its proper divisors, so for example $6=1+2+3$ and $28=1+2+4+7+14$ are both perfect.) Euclid’s Book IX, Proposition 36 tells us that if the quantity $2^{n} – 1$ is prime, then $2^{n-1} \times (2^{n} – 1)$ is perfect. It turns out that every even perfect number has this form (it is still not known whether an odd perfect number exists).
Descartes and Fermat studied perfect numbers, as did the French cleric and polymath Marin Mersenne (1588-1648), who corresponded with both of them. In 1644, Mersenne published a list of all the perfect numbers that had been known before him. There were eight, the last one being constructed, following Euclid’s proposition, with $(2^{31}-1)2^{30}=2,305,843,008,139,952,128$ (roughly 2.3 million trillion). To extend the list Mersenne looked for even larger prime numbers of the form $2^n-1$. It turns out if $2^n – 1$ is prime, $n$ must also be prime, although the converse is not true: there are cases where $n$ is prime and $2^n – 1$ is not, e.g. $2^{11}-1=89\times 23$. Numbers of the form $2^p – 1$ (where $p$ is prime) are now called Mersenne numbers.
Mersenne claimed to have used this method to discover three more perfect numbers, but of his candidates only one is actually perfect. Nevertheless, his tactic of searching for primes within the Mersenne numbers was a good one, better in fact than he knew. Back in 1644, Mersenne commented on the difficulty of recognizing “whether given numbers consisting of 15 or 20 digits are prime or not, since not even an entire century is sufficient for this investigation, in any way known up to now.” But subsequent progress in number theory made it much easier to determine whether a Mersenne number is prime or not than to do it for a random integer of the same size, and almost all the largest primes discovered have been of this type.
The 52nd and most recent Mersenne prime, computed with $p=136,279,841$, has over 41 million digits. (You can watch them stream by, at the rate of 100,000 per second, in this video.) Ben Brasch of the Washington Post covered the discovery on October 23 under the headline: “One year, 41 million digits: How he found the largest known prime number.” Brasch focuses on Luke Durant, the 36-year-old chip design veteran who used GIMPS software to farm out the search for a new Mersenne prime to publicly accessible servers. “I was able to do it just by using big tech’s leftovers,” Durant told the Post.
He ended up using servers from 24 different data centers. This kind of distributed computing had been used in the discovery of the last 17 largest primes; the difference is that Durant organized a shift from ordinary computer CPUs to GPUs (graphics processing units), which worked almost an order of magnitude faster. This allowed the collective to find a prime over 16 million digits longer than the last largest one, detected with CPUs back in 2018.
Durant—who put $2 million of his own fortune into the project—frames it as a way to show people what they can accomplish if they work together. Brasch quotes him: “The scale of computing available in the cloud, it’s nearly unfathomable. … [W]e have these incredible systems, so let’s figure out how to best use them.”
Concepts vs. practice in Kindergarten math education.
High school Algebra has a high failure rate, a phenomenon that has been in the news for the last couple of years at least. (See, for example, items in this column from May and November 2023.) Sarah Schwartz covered a new initiative to remedy this problem in Education Week on October 9. Her article is headlined: “Can Kindergarten Math Lay the Foundation for Algebra? New Study Aims to Find Out.” The study in question tests lessons from Project LEAP, a curriculum for elementary school students that hopes to encourage “algebraic thinking.”
Maria Blanton is a senior scientist at TERC, the non-profit behind Project LEAP. She explained the logic underlying the project to Schwartz: In elementary school, students are taught to work with numbers. But then, “they’re dropped into a class where, all of the [sic] sudden, the numbers become letters.” The solution, Project LEAP believes, is to instill “algebraic thinking skills” from the start. Schwartz cites a 2001 article in the Journal of Educational Psychology with the title: “Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process.”
An example can help bring the project into focus: children’s understanding of the equals sign. We have all heard $7-4=3$ read as “seven take away four leaves three.” But young children can misconstrue this phrasing to mean that “$=$” is an operational symbol meaning, as Schwartz puts it, “the answer comes next.” Students might then get confused by statements like $8=8$, as an elementary school math teaching specialist told Schwartz.
Another example came from Blanton. Early on, students learn which numbers are even and which are odd. The early algebra approach would build on this experience, “allowing them to make generalizations about how odd and even numbers operate.” These ideas would be made concrete, for example by having lined-up blocks represent numbers. Then an even number would be made up of pairs of blocks, but an odd number would have a singleton. Students can see how two odd numbers add up to give an even, since the two singletons would form a pair.
According to Blanton, a previous grades 3-5 study showed the project LEAP students as scoring better than the controls, but “their overall ability was still low.” The hope is that earlier exposure will lead to further improvement. Another study examining the effectiveness of Project LEAP in grades K-2 started this academic year and continues through May 2027. In this experiment, Project LEAP lessons are implemented in 41 schools, while a control group of classes sticks to their usual curriculum.
This project is an implementation at the most elementary level of what I believe to be the way mathematics develops: an iterative process in which the systematic investigation of mathematical phenomena leads to the discovery of new ones, which then get systematically investigated: the procedural leads to the conceptual, and vice versa.
“Why physics is unreasonably good at creating new math.”
This is the title of a Nautilus article by Ananyo Bhattacharya, reposted on the website Big Think. The headline evokes Eugene Wigner’s famous article “The unreasonable effectiveness of mathematics in the natural sciences.” Bhattacharya flips this around, writing that “the tables have turned. Now insights and intuitions from physics are unexpectedly leading to breakthroughs in mathematics.” This new development seems even more “unreasonable,” and we are led to examine how it has come about.
For one possible explanation Bhattacharya quotes Timothy Gowers (Collège de France, Fields Medalist): “Physicists are much less concerned than mathematicians about rigorous proofs.” This presumably allows them, unconstrained by logical scruples, to flit rapidly over the mathematical landscape and spot new phenomena which mathematicians can then laboriously investigate. More seriously, Bhattacharya reminds us that physics inspiring mathematics was the standard pattern for centuries. Archimedes used the laws of mechanics to solve math problems; for example, relating the volume of a sphere to the volume of a cylinder. Newton discovered (or invented) calculus while trying to understand gravity.
But in the middle of the 20th century, the two disciplines grew apart. Bhattacharya credits the late Michael Atiyah (1929-2019) with instigating their reconciliation. The Hitchin article he cites traces this back to the remarkable physical consequences of the 1963 Atiyah-Singer Index Theorem. Another factor was Atiyah’s long friendship and collaboration with the physicist and Fields Medalist Edward Witten, who appears in Bhattacharya’s account and who is arguably the single most important figure in the whole story. Among other contributions, Witten pioneered string theory, which is still controversial in physics but has generated deep developments in mathematics. As an illustration Bhattacharya shows this picture of (a 2-dimensional section of) a Calabi-Yau manifold. These complicated 6-dimensional spaces, which came up in string theory, have been the focus of intensive research in topology and differential geometry.
Getting back to the original question, Bhattacharya elicits a sensible explanation from the mathematical physicist Yang-Hui He (London Institute for Mathematical Sciences): of all the patterns and structures that mathematicians can study, “the ones which come from reality are ones which we have an intuition about at some level.”
—Tony Phillips, Stony Brook University