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Tony Phillips' Take on Math in the Media A monthly survey of math news |
A press release from GIMPS (the Great Internet Mersenne Prime Search) was picked up by the BBC on January 20, 2016: "Largest known prime number discovered in Missouri," a title which led Evelyn Lamb to comment on Slate (January 22) that it sounds "a bit like this new prime number ... was found in the middle of some road." The BBC posting shows the beginning of a scroll-through of all of M74207281, as this number ($2^{74207281}-1$) is called, with perky musical accompaniment. There's twenty minutes' worth; the complete file is available from the press release: click on "22,338,618 digits" to savor it at your leisure.
On Slate the coverage is more extensive, and quite enthusiastic. "Here's What's So Exciting About the New Largest Prime Number Mathamaticians Discovered" is the link from their fron page. As Lamb tells us, "GIMPS ... is a large distributed computing project in which volunteers run software to search for prime numbers." She tells us what Mersenne primes are ("The M in GIMPS and in M74207281 stands for Marin Mersenne, a 17th-century French friar [but much more -TP] who studied the numbers that bear his name. Mersenne numbers are 1 less than a power of 2. Mersenne primes, logically enough, are Mersenne numbers that are also prime. The number 3 is a Mersenne prime because it's one less than $2^2$ which is 4") and what we've lost by concentrating on the Mersennes: she uses the prime number theorem to estimate that "There are about $10^{22,338,610}$ primes less than M74207281, and approximately all of them are between it and the next-largest known prime." She also explains the reason for that concentration: there is a special primality test "that can determine whether a number is prime without actually factoring it," that only works for Mersenne numbers. For a lucid description of this test, the Lucas-Lehmer algorithm, listen to Matt Parker on Numberphile.
The story also ran in the New York Times on January 22. Kenneth Chang's "New Largest Prime Number? It's Really, Really Long" starts: "The largest known prime number, newly discovered, is almost five million digits longer than the previous record-holder." Chang quotes George Woltman, who founded the GIMPS project after he retired 20 years ago: "I've always been interested in prime numbers. I had a lot of time on my hands." And he spoke with Curtis Cooper (University of Central Missouri) whose computer found M74207281: "one of the early enthusiasts, joining Gimps in 1997. He has the program currently installed on 800 PCs on the university's two campuses. Dr. Cooper does research in the mathematical realm of number theory and teaches computer science classes. 'This kind of marries the two fields together,' he said."
A nice philosophical point: Chang also tells us that the UCM computer (PC No. 5 in Room 143) "churned for 31 days before completing its calculation that $2^{74207281}-1$ is a prime," and reporting the result, on September 17, 2015. But because of a "glitch on the server" no human being was informed. The official discovery date is January 7, 2016 when the message was picked up during routine maintenance.
"Undecidability of the spectral gap," by Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf, was published in Nature, December 10, 2015. From their abstract: "The spectral gap --the energy difference between the ground state and first excited state of a system-- is central to quantum many-body physics. Many challenging open problems ... are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable."
"The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'."
A schematic representation of the physical system built from the Robinson tiling. Image from Nature 528 207-211, used with permission
An interesting mathematical aspect of the construction is authors' use of planar tilings as the link between Turing machines and physics. On the one hand they were able to translate the geometry of a tiling into the configuration of an (ideal) physical system for which a problem, equivalent to the mass-gap problem, was also equivalent to the completion problem for the tiling: can a given partial tiling can be extended to the entire plane? On the other hand they exploited work done by Hao Wang, Robert Berger and Raphael Robinson in the 1960s and 70s on the decidability of of the completion problem. In particular they cite Robinson's 1971 article (Inventiones Math. 12 177-209), which gives a specific way of representing a Turing machine by a collection of tiles with matching rules. Robinson states: "We see that the tiling of the plane can be completed with the tiles mentioned, after the initial tile has been laid down, if and only if the Turing machine never halts." Since Turing had proved that the halting problem is undecidable, the same must hold for the completion problem of the corresponding tiling.
The six tiles invented by Raphael Robinson, which can tile the plane but only aperiodically. Because these tiles are essentially square-shaped (unlike Penrose tiles), one row can be interpreted as the state and input of a Turing machine; then the next row, determined by the matching rules, givs the next state. Public domain image from Wikipedia.
The trace of a Robinson tiling. The brown marks on the tiles fit together to give the blue and brown interlocking squares. (More detail in this image by Chaim Goodman-Strauss). Since smaller squares group four by four with larger ones, this pattern cannot be periodic. In the image from Nature, only the blue squares (odd-numbered sizes) are shown.
The article was highlighted in a "News and Views" piece by Davide Castelvecchi in the same issue of Nature: "Paradox at the heart of mathematics makes physics problem unanswerable," showing photographs of Gödel and Turing. "Gödel's incompleteness theorems are connected to unsolvable calculations in quantum physics."
If we really want to know, there's a piece in the Careers section of Nature (December 10, 2015) that spells it out. Chris Woolston, a freelance writer, starts with "These days in science, there's no escape from maths in any scientific discipline, even in one like marine biology, historically lighter on sums than, say, molecular biology or quantitative genetics. But nobody should let maths jitters deter them if their call is to study ocean life." He discusses this point with some active marine biologists. A sample:
Woolston lists some "Resources for mathophobes" including the website StackExchange, Vi Hart and free online audio-visual tutorials from the Elementary Maths for Biologists course, Cambridge, UK. In case none of those helps, he ends the list with a link to E.O. Wilson's Great Scientist $\neq$ good at math from the Wall Street Journal and the quote: "Many of the most successful scientists in the world today are mathematically no more than semiliterate."
Tony Phillips
Stony Brook University
tony at math.sunysb.edu