For over 7 years, mathematicians David H. Bailey and Jonathan M. Borwein have published essays, new items, quotations and book reviews (236 posts in total). Our posts have included:

Notices of new mathematical discoveries: see Sphere packing problem solved in 8 and 24 dimensions and Unexpected pattern found in prime number digits. Descriptions of new developments in the larger arena of modern science: see Space exploration: The future is now and Gravitational waves detected, as predicted by Einstein’s mathematics. Discussions of scientific controversies: see How likelyContinue reading Introducing the Math Scholar blog

]]>For over 7 years, mathematicians David H. Bailey and Jonathan M. Borwein have published essays, new items, quotations and book reviews (236 posts in total). Our posts have included:

- Notices of new mathematical discoveries: see Sphere packing problem solved in 8 and 24 dimensions and Unexpected pattern found in prime number digits.
- Descriptions of new developments in the larger arena of modern science: see Space exploration: The future is now and Gravitational waves detected, as predicted by Einstein’s mathematics.
- Discussions of scientific controversies: see How likely is it that scientists are engaged in a conspiracy? and Why E.O. Wilson is wrong.
- Fermi’s paradox: see Where is ET? Fermi’s paradox turns 65 and Desperately seeking ET: Fermi’s paradox turns 65, Part II.
- Artificial intelligence: see IBM’s “Watson” victorious: Our new computer overlords? and What does Watson’s victory really mean?.
- Creationism: see How certain are scientists that the earth is many millions of years old? and Are there “missing links” in the human family tree?.
- Pi: see Pi Day 2016 and More mathematics (and Pi) in the media.

Sadly, Jonathan Borwein passed away on 2 August 2016. As I wrote the day of his passing,

Jon’s passing is an incalculable loss to the field of mathematics in general, and to experimental mathematics in particular. Jon is arguably the world’s leading researcher in the field of experimental mathematics, and his loss will be very deeply felt. We will be reading his papers and following his example for decades to come.

In the wake of Jon’s passing, it seems appropriate to end the Math Drudge blog. Thus no new posts will be made to this site.

Readers are instead referred to a new blog, the Math Scholar blog. In this forum, myself and some other guest writers will explore many of the same themes mentioned above, as well as some new topics that emerge in our ever-changing world.

Comments are welcome. Please contact the present author.

]]>What can one say about Jon’s professional accomplishments? Adjectives such as “profound,” “vast” and “far-ranging” don’t really do justice to his work, the sheer volume of which is astounding: 388 published journal articles, plus another 103

Continue reading Jonathan Borwein dies at 65

]]>What can one say about Jon’s professional accomplishments? Adjectives such as “profound,” “vast” and “far-ranging” don’t really do justice to his work, the sheer volume of which is astounding: 388 published journal articles, plus another 103 articles in refereed or invited conference proceedings (according to his CV, dated one day before his death). The ISI Web of Knowledge lists 6,593 citations from 351 items; one paper has been cited 666 times. The Google Citation Tracker finds over 22,048 citations.

But volume is not the only remarkable feature of Jon’s work. Another is the amazing span of his work. In an era when academic researchers in general, and mathematicians in particular, focus ever more tightly on a single specialty, Jon ranged far and wide, with significant work in pure mathematics, applied mathematics, optimization theory, computer science, mathematical finance, and, of course, experimental mathematics, in which he has been arguably the world’s premier authority.

Unlike many in the field, Jon tried at every turn to do research that is accessible, and to highlight aspects of his and others’ work that a broad audience (including both researchers and the lay public) could appreciate. This was, in part, behind his long-running interest in Pi, and in the computation and analysis of Pi — this topic, like numerous others he has studied, is one whose wonder and delight can be shared with millions.

This desire to share mathematics and science with the outside world led to his writing numerous articles on mathematics, science and society for the Math Drudge blog, the Conversation and the Huffington Post. He was not required to do this, nor, frankly, is such writing counted for professional prestige; instead he did it to share the facts, discoveries and wonder of modern science with the rest of the world.

Jon was a mentor par excellence, having guided 30 graduate students and 42 post-doctoral scholars. Working with Jon is not easy — he is a demanding colleague (as the present author will attest), but for those willing to apply themselves, the rewards have been great, as they become first-hand partners in ground-breaking work.

There is much, much more that could be mentioned, including his tireless and often thankless service on numerous committees and organizational boards, including Governor at large of the Mathematical Association of America (2004–07), President of the Canadian Mathematical Society (2000–02), Chair of the Canadian National Science Library Advisory Board (2000–2003) and Chair of the Scientific Advisory Committee of the Australian Mathematical Sciences Institute (AMSI).

But Jon was more than a scholar. He was a devoted husband and father. He and Judi have been married for nearly 40 years, and they have three lovely and accomplished daughters. They have endured some incredible hardships, but Jon has made some equally incredible sacrifices on their behalf. Jon has also been devoted to his own father and mother, often collaborating on research work with his father David Borwein (also a well-known mathematician), and following the work of his mother, a scholar in her own right.

I myself am at a loss of what to say at Jon’s passing. What can I say? I have collaborated with Jon for over 31 years, with over 80 papers and five books with Jon as a co-author. Thus my personal debt to Jon is truly enormous. My work will forever be connected with (and certainly subservient to) that of Jon’s. I am humbled beyond measure and grieve deeply at his passing.

Jon’s passing is an incalculable loss to the field of mathematics in general, and to experimental mathematics in particular. Jon is arguably the world’s leading researcher in the field of experimental mathematics, and his loss will be very deeply felt. We will be reading his papers and following his example for decades to come.

BORWEIN, Jonathan Michael, August 2, 2016 at age 65. Jon Borwein, Laureate Professor at the University of Newcastle, FRSC, FAAAS, FBAS, FAustMS, FAA, FAMS, FRSNSW passed away suddenly at 12:32 a.m. on August 2 during a stint as Distinguished Scholar in Residence at Western University in London, Ontario.

An innovative and prolific mathematician of international renown, he is survived by his parents, Bessie and David Borwein of London; his adoring wife Judith, three daughters Naomi, Rachel, and Tova; five grandchildren Jakob Joseph, Noah Erasmus, Skye, Zoe and Taj; siblings Sarah and Peter; sister-in-law Jennifer Moore.

The funeral will take place at Logan Funeral Home, 371 Dundas Street, London, Ontario at 1:00 pm on Wednesday August 10, 2016 with a visitation period from noon to 1:00 pm. In lieu of flowers, friends and colleagues may wish to donate to a scholarship fund being set up in Jonathan’s name. For information on how to donate contact Judith.Borwein@gmail.com.

]]>The present authors are not the best examples of this, because neither is very good at musical performance, although both have an abiding interest in listening to music. One of us listens to an eclectic collection of mostly modern music while

Continue reading Why are so many mathematicians also musicians?

]]>As the present authors will readily attest, introducing oneself as a mathematician is generally not an effective way to start a social conversation. But, as Cambridge mathematician Tim Gowers explains, there is a “miracle cure”: just explain that you, as well as many other mathematicians, are also a musician or at least are deeply interested in music.

The present authors are not the best examples of this, because neither is very good at musical performance, although both have an abiding interest in listening to music. One of us listens to an eclectic collection of mostly modern music while he works (rock, jazz, classical, singer-song-writer, show tunes, alternative, folk, country, adult contemporary and world music to name a few — plus an Apple music subscription). The other one of us has a large collection of classical music, including the entire works of Bach (nearly 200 hours total), on his iPhone, all of which he has listened to many times.

Perhaps the best real-life example of a mathematician-musician was Albert Einstein, who, as many who knew him personally would attest, was also an accomplished pianist and violinist. His second wife Elsa told of how Albert, while during deep concentration on a mathematical problem, would sit down at the piano and play for a while; after one two-week period, interspersed with random piano playing, Einstein emerged the first working draft of general relativity. He once said, “If … I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.”

So why is it that a remarkable number of professional mathematicians are also into music? Are the two disciplines so similar? Or is there a genetic link? Or is it simply that both mathematicians and musicians are likely to have been raised in households where mathematics, music and other scholarly, artsy subjects were valued, and where the mathematicians and/or musicians were encouraged on by eager parents? Good questions.

There does seem to be a credible connection between the sort of mental gymnastics done by a mathematician and by a musician. Additionally, current work on synesthesia supports the notion that creativity is enhanced by a mixing of the senses.

To begin with, there are well-known mathematical relationships between the pitch of various notes on the musical keyboard. An octave is separated by a factor of two; a fifth interval (say C to G) by the ratio 3/2, and two adjacent notes on the keyboard are separated by the twelfth root of two = 1.059463…

But beyond mere analysis of pitches, it is clear that the deeper world of musical syntax and structure is akin to the sorts of sophisticated structures, syntax and regularities that are part and parcel of mathematical thinking. Mozart is well-known to have written music that is both beautiful and structurally impeccable. As Salieri explained in the musical *Amadeus*, “Displace one note and there would be diminishment. Displace one phrase and the structure would fall.”

Arguably the most “mathematical” composer of all was Johann Sebastian Bach, who was a master of counterpoint and polyphony. His work typically starts with a fairly simple theme (in the case of the monumental Brandenburg Concerto #5, it is merely a simple four-note pattern), then combines the theme with offsets, much as a chorus does when singing “rounds,” in ever-more-elaborate ways, thus producing an often stunning result.

One example, which can visualized using a very nice online tool, is Bach’s “Great” Fugue in G minor (BWV 542). The theme is introduced immediately, and then developed into countless polyphonic variations. An even better example is Bach’s Fugue in E minor (BWV 548), known as the “Wedge” fugue, so named for a strong theme that develops as an expanding sequence of notes in the shape of a wedge, quite obvious in the printed score (see sidebar), and then is repeated in countless polyphonic variations, all connected in a sophisticated high-level structure.

With these connections between mathematics and music, it was perhaps inevitable that both mathematicians and musicians would turn to computers. Thus the field of computer music was born. Numerous tools have been developed to assist in this task; indeed, many modern-day musicians, covering a wide range of specialties, now utilize computers in their work.

Some of the more interesting work in this area is to program computers to actually compose music. David Cope, for instance, has written computer programs that can analyze a corpus of music, say by a particular composer, and then create new works in a similar style. He was most successful in replicating and producing variations of the music of Bach and Mozart, which is perhaps not surprising given the highly mathematical structures used by these composers. One of Cope’s programs, known as “Experiments in Musical Intelligence,” created what he dubbed “Mozart’s 42nd Symphony.”

So to return to one of our original questions, is there a link between mathematical talent and musical talent? There are certainly many musicians who didn’t make it past algebra in school, and there are certainly many mathematicians who cannot carry a tune. Furthermore, it is not as easy as one might think to scientifically test such a proposition. As Tim Gowers observes,

[I]f you want to show that professional mathematicians are on average better at music than other people, then you have to decide quite carefully who those “other people” are. You might expect that the kind of person who becomes a professional mathematician is much more likely than average to come from the kind of family that would consider music to be an important part of a child’s education, so for that reason alone one would expect at least some “background correlation” between the two. … Identifying and controlling for these kinds of effects is difficult, and as far as I know, … there has been no truly convincing study that has shown that musical ability enhances mathematical ability or vice versa.

In short, while it is problematic to claim any kind of innate link between mathematical ability and musical ability, it is clear that the two disciplines have a deep commonality.

One of us (Borwein) has taught many students who were vacillating between musical, medical, and academic careers. And at many mathematical conferences, entertainment is provided by international-level pianists or violinists whose day job is mathematics.

What’s more, in the era of modern high-performance computing, the future may bring the two disciplines together in ways that we can scarcely imagine at the present time. May your mathematical future also be a musical one!

]]>However, many have been concerned lately that the glory days of space exploration are behind us. The Apollo missions ended 44 years ago, and still we have not returned to the Moon. Our current Mars missions are only modestly more sophisticated than earlier missions. And

Continue reading Space exploration: The future is now

]]>From the dawn of civilization, humans have dreamed of exploring the cosmos. To date, we have launched over 60 successful missions to the Moon (including six that landed on the Moon with humans), 17 successful missions to Mars, 13 missions to the outer solar system, and five that have left the solar system.

However, many have been concerned lately that the glory days of space exploration are behind us. The Apollo missions ended 44 years ago, and still we have not returned to the Moon. Our current Mars missions are only modestly more sophisticated than earlier missions. And futuristic dreams of humans traveling to the planets and to the cosmos have remained decades if not centuries away.

But within the past year or so, this situation seems to be changing. Perhaps it has been inspired by a string of highly successful Hollywood movies, including Gravity, Interstellar, The Martian and Star Wars: The Force Awakens. Perhaps it stems in part from the surprising success of private firms such as SpaceX and Blue Origin. Or perhaps it is simply the insatiable curiosity and wanderlust that is so deeply ingrained into our species via evolution.

Arguably the most daring plan to date is the Breakthrough Starshot project that was announced on 12 April 2016 by Russian billionaire Yuri Milner, with backing from physicist Stephen Hawking and Facebook founder Mark Zuckerberg.

Milner proposes to send a fleet of “nanocraft” to explore Alpha Centauri and its planets — thousands of credit-card-sized spacecraft (to increase the chances that at least some will survive the journey), quickly accelerated to 20% of the speed of light by giant light sails powered by laser beams from a kilometer-square array of earth-bound lasers.

Upon arrival in the Alpha Centauri system in 20 years or so, the nanocraft will send back photos and other data via laser beams, which will arrive at Earth four years later. Needless to say, this vision presents daunting technical hurdles, including:

- Fabricating diode lasers, megapixel cameras, computer processors and batteries for the nanocraft, together weighing less than one gram and able to survive 20 years of exposure to interstellar dust and cosmic rays.
- Maintaining integrity of the light sail while it and its nanocraft are being accelerated by lasers.
- Producing sufficient laser power and maintaining the focus of the laser array.
- Detecting the images and data that are sent back to Earth.

For additional details, see the Scientific American report.

An entirely different, but comparably ambitious, proposal to study extraterrestrial civilizations is to use the Sun as a gravitational lens. SETI pioneer Frank Drake, among others, proposes sending spacecraft outside the solar system to the focal point of the Sun’s gravitational field, which, by principles of General Relativity, can then see enormously magnified images and even microwave transmissions coming from a distant star system.

An equally significant development is the resurgence in interest for humans not only to visit Mars but also to take up residence and ultimately form an independent colony.

The Mars Society observes that a round-trip journey to Mars is possible by manufacturing fuel for the return trip *in situ* on Mars (otherwise transporting fuel to Mars for the return trip is 90% of the outbound payload). In particular, they note that CO_{2} extracted from Martian atmosphere and H_{2} produced from Martian ice by electrolysis can be combined to form fuel by the *exothermic* reaction 3 CO_{2} + 6 H_{2} → CH_{4} + 2 CO + 4 H_{2}O. A fully fueled and tested lift-off vehicle could be ready and waiting on Mars before the astronauts leave Earth.

Mars One, an organization founded in 2012, proposes to send humans to Mars by 2027 and establish a permanent colony there, to be funded in part by a reality TV show. Over 200,000 persons responded to their 2013 call for interest; this list has now been narrowed down to 100.

Tesla and SpaceX founder Elon Musk has also been formulating plans for a Mars colony, which he has promised to announce soon. His project reportedly will be known as the Mars Colonial Transporter, to be powered by a large version of the Raptor rocket engine, specifically designed for the exploration and colonization of Mars.

Looking a bit further to the future, advanced propulsion systems will be required if more than a handful of people are to reach Mars or beyond. NASA has been exploring several concepts in this direction, including:

- Ion propulsion: A high-energy electron collides with a xenon atom, releasing electrons, and the charged atom is then discharged at high speed (up to 150,000 kph).
- High-power electric propulsion: This is like ion propulsion, except that the xenon ions are produced by a combination of microwave and magnetic fields, using a process called electron cyclotron resonance.
- Fusion-driven rocket: A fusion energy source releases its energy directly into the propellant, without converting to electricity; the propellant is rapidly heated and accelerated to high exhaust velocity (roughly 100,000 kph) with no physical interaction with the spacecraft, thus avoiding deterioration.

These developments have clear implications for Fermi’s paradox, that decades-old unsolved conundrum of why, given that an extraterrestrial civilization could explore the Milky Way in a million years or so (an eyeblink in cosmic time), do we not see evidence of even a single society?

Numerous scientists have examined Fermi’s paradox in detail and, as we wrote earlier, have proposed various explanations, such as:

- They exist, but are too far away.
- They are under strict orders not to disclose their existence.
- They exist, but have lost interest in communication and exploration.
- They are calling, but we do not recognize the signal.
- Civilizations like us invariably self-destruct.
- We are alone, at least within the Milky Way if not beyond.

All of these explanations have very reasonable rejoinders. Items 2 and 3 (and several other similar proposed explanations) fall prey to a diversity argument — in a vast galactic ensemble it is hardly credible that *every* individual in *every* civilization forever lacks interest in communication and exploration, nor is it credible that some galactic society ban is absolutely 100% effective (note that once a signal has been sent, it cannot be called back by any known law of physics). Item 4 does not seem credible, since it is very reasonable to assume that at least some communications are being sent to planets such as Earth in a form that we could readily recognize, and, as before, it is not credible that a ban on such targeted communication could be absolutely 100% effective. Item #6 (we are alone) seems incredible in light of the thousands of recently discovered extrasolar planets, many in the habitable zone.

With regards to Item 1 (they exist, but are too far away), it is clear that the many exciting new developments in space exploration seriously draw into question the presumed technical impossibility of exploring the cosmos. For example, a fleet of “von Neumann probes” could travel to distant stars, make additional copies of themselves (using the latest software beamed from the home planet) and launch to yet more distant stars. Analyses of this scheme show the entire Milky Way could be explored in a one million years or so. And keep in mind that any other society is, almost certainly, many thousands or millions of years more advanced, so cost and distance cannot be insuperable obstacles.

These developments also draw Item 5 into question (civilizations like us invariably self-destruct). After all, we have survived 200 years of technological adolescence and have not yet destroyed ourselves. And if any of the current exploration and colonization plans work out, then the long-term survival of our species will be immune to possible calamities on Earth. Within a decade, we will become a multi-planet species, and within a century we very likely will be a multi-solar-system species.

So what is the answer to Fermi’s paradox? Good question! We humans don’t know.

[Added 27 Apr 2016:] Elon Musk’s SpaceX has announced that it plans to land an unmanned spacecraft on Mars by 2018, with some help by NASA. For details, see this Washington Post article.

]]>In the 17th century, Johannes Kepler conjectured that the most space-efficient way to pack spheres is to arrange them in the usual way that we see oranges stacked in the grocery store. However, this conjecture stubbornly resisted proof until 1998, when University of Pittsburgh mathematician Thomas Hales, assisted by Samuel Ferguson (son of mathematician-sculptor Helaman Ferguson), completed a 250-page proof, supplemented by 3 Gbyte of computer output.

However, some mathematicians were not satisfied with Hales’ proof, as it relied so heavily on computation. So Hales embarked on project Flyspeck, which was to construct a

Continue reading Sphere packing problem solved in 8 and 24 dimensions

]]>In the 17th century, Johannes Kepler conjectured that the most space-efficient way to pack spheres is to arrange them in the usual way that we see oranges stacked in the grocery store. However, this conjecture stubbornly resisted proof until 1998, when University of Pittsburgh mathematician Thomas Hales, assisted by Samuel Ferguson (son of mathematician-sculptor Helaman Ferguson), completed a 250-page proof, supplemented by 3 Gbyte of computer output.

However, some mathematicians were not satisfied with Hales’ proof, as it relied so heavily on computation. So Hales embarked on project Flyspeck, which was to construct a completely detailed, machine-checkable formal proof of the conjecture. Finally, on 10 August 2014, Hales announced that the project was complete.

Hales’ proof of the Kepler conjecture was limited to three dimensions. Even prior to Hales’ original proof, mathematicians had explored generalizations of Kepler’s conjecture in higher dimensions.

So the mathematical community reacted with considerable interest when, in March 2016, Ukrainian mathematician Maryna Viazovska posted a solution, with proof, of the sphere packing problem in dimension 8, and, just a week later, also in dimension 24.

Based on computational analysis, mathematicians for some time had suspected that the optimal packing in dimension 8 was the E8 lattice, one of the fundamental objects in Lie algebras and groups, and that the optimal packing in dimension 24 was the Leech lattice. But until Viazovska’s papers, there was no proof.

The E8 lattice has been explored by researchers for many reasons. It also has connections to string theory in physics. The Leech lattice has been employed in coding theory, because it is capable of detecting up to four errors in a 24-bit word and correcting up to three errors.

Viazovska’s proof for dimension 8 employed the theory of modular forms, which are complex analytic functions satisfying a certain type of functional equation with respect to the actions of the modular group. Her proof was praised by Princeton mathematician Peter Sarnak, who said, “It’s stunningly simple, as all great things are. … You just start reading the paper and you know this is correct.”

Just one week after her original paper was posted on arxiv.org, she posted a second paper, co-authored with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Danylo Radchenko, that employed a similar approach to solve the dimension 24 case.

Additional background and details on Viazovska’s work can be read in a nice article by Erica Klarreich in Quanta Magazine, on which part of the above post was based. Viazovska’s technical paper for dimension 8 is available here, and her paper for dimension 24, co-authored with the four other mathematicians, is available here.

]]>The seventeen thought-provoking and engaging essays in this collection present readers with a wide range of diverse perspectives on the ontology of mathematics. The essays address such questions as: What kind of things are mathematical objects? What kinds of assertions do mathematical statements make? How do people think and speak about mathematics? How does society use mathematics? How have our answers to these questions changed over the last two millennia, and how might they change

Continue reading New book on the ontology of mathematics

]]>The seventeen thought-provoking and engaging essays in this collection present readers with a wide range of diverse perspectives on the ontology of mathematics. The essays address such questions as: What kind of things are mathematical objects? What kinds of assertions do mathematical statements make? How do people think and speak about mathematics? How does society use mathematics? How have our answers to these questions changed over the last two millennia, and how might they change again in the future? The authors include mathematicians, philosophers, computer scientists, cognitive psychologists, sociologists, educators and mathematical historians; each brings their own expertise and insights to the discussion.

The present authors contributed the article “Experimental computation as an ontological game changer: The impact of modern mathematical computation tools on the ontology of mathematics.” We point out that computational tools, and the hardware they run on, are becoming so powerful that in many cases they constitute a mode of research equally compelling to the traditional mode of axiomatic proof. At the least, these tools raise fundamental questions about what is the nature of secure mathematical knowledge and how we discover and confirm that knowledge.

Other essays in the collection include the following:

- Ursula Martin and Alison Pease, “Hardy, Littlewood and polymath”
- Philip J. Davis, “Mathematical products”
- Ernest Davis, “How should robots think about space?”
- David Berlinski, Mathematics and its applications”
- Jody Azzouni, “Nominalism, the nonexistence of mathematical objects”
- Donald Gillies, “An Aristotelian approach to mathematical ontology”
- Jesper Lutzen, “Let G be a group”
- John Stillwell, “From the continuum to large cardinals”
- Jeremy Gray, “Mathematics at infinity”
- Jeremy Avigad, “Mathematics and language”
- Micah T. Ross, “The linguistic status of mathematics”
- Kay L. O’Halloran, “Mathematics as multimodal semiosis”
- Steven T. Piantadosi, “Problems in philosophy of mathematics: A view from cognitive science”
- Lance J. Rips, “Beliefs about the nature of numbers”
- Nathalie Sinclair, “What kind of thing might number become?”
- Helen Verran, “Enumerated entities in public policy and governance”

Additional details, previews and other material are available at the Springer site.

]]>Fermat’s Last Theorem was first conjectured in 1637 by Pierre de Fermat in 1637, in a cryptic annotated marginal note that Fermat wrote in his copy of Diophantus’ Arithmetica. For 358 years, the problem tantalized generations of mathematicians, who sought in vain for a

Continue reading Andrew Wiles wins the Abel Prize

]]>Fermat’s Last Theorem was first conjectured in 1637 by Pierre de Fermat in 1637, in a cryptic annotated marginal note that Fermat wrote in his copy of Diophantus’ Arithmetica. For 358 years, the problem tantalized generations of mathematicians, who sought in vain for a valid proof. In the 1995 edition of the Guiness Book of World Records, it was cited as the world’s most difficult mathematical problem, in part because of the large number of unsuccessful proofs through the ages. Some of these proofs were foolish, but others helped build modern number theory.

In the mid-1970s, Wiles had trained under Cambridge University based Australian mathematician John Coates, who had recently returned to England from teaching at Stanford University. They studied the arithmetic of elliptic curves, using the various methods, including Iwasawa theory. (A personal note: Prior to working with Wiles, in 1972 John Coates taught one of the present authors (Bailey) a course in Algebra at Stanford.)

The proof of Fermat’s Last Theorem had its origin in a series of results in the 1980s by Gerhard Frey of the University of Duisburg-Essen, Jean-Pierre Serre of the Centre National de la Recherché Scientifique College de France, and Ken Ribet of the University of California, Berkeley. From their work, it became clear that Fermat’s Last Theorem might be proven as a consequence of a limited form of the Taniyama-Shimura-Weil conjecture, which is now known as the modularity theorem.

Wiles, who had been fascinated by Fermat’s Last Theorem since childhood, decided to pursue a proof. After working in secret for several years on the project, on 24 June 1993, he announced his result in a lecture at Cambridge University.

Alas, a few months later a flaw was uncovered in his proof. Finally, in 1995, a full corrected proof was published in Annals of Mathematics, with one of the two final papers co-authored with Richard Taylor. The proof has stood the test of time — 20 years later no flaw has been uncovered.

We add our congratulations to Wiles for his landmark achievement, and hope that his example will inspire many other mathematicians to pursue lines of research traditionally thought to be “too difficult.” A brief outline of Andrew Wiles’ proof is presented in the Wikipedia article on the topic. Additional information on both Wiles and the Abel Prize, modeled on the Nobel prize, is available at the Norwegian Academy’s website, and in well-written articles in New Scientist and Nature.

]]>In base ten digits, for example, all primes greater than 5 end in 1, 3, 7 or 9, since otherwise they would be divisible by 2 or 5. Under the common assumption that prime numbers resemble good pseudorandom number generators, a prime ending in 1, for instance, should

Continue reading Unexpected pattern found in prime number digits

]]>In base ten digits, for example, all primes greater than 5 end in 1, 3, 7 or 9, since otherwise they would be divisible by 2 or 5. Under the common assumption that prime numbers resemble good pseudorandom number generators, a prime ending in 1, for instance, should be followed by a prime ending in 1, 3, 7 or 9 with equal probability, i.e., 25%. Thus, in the first 100,000,000 primes, one would expect each of the 16 pairs “1, 1”, “1, 3”, “1, 7”, “1, 9”, … “9, 1”, “9, 3”, “9, 7”, “9, 9” to appear 6,250,000 times.

Instead, Lemke Oliver and Soundararajan found the sixteen cases as follows:

1, 1 | 4,623,042 | 7, 1 | 6,373,981 |

1, 3 | 7,429,438 | 7, 3 | 6,755,195 |

1, 7 | 7,504,612 | 7, 7 | 4,439,355 |

1, 9 | 5,442,345 | 7, 9 | 7,431,870 |

3, 1 | 6,010,982 | 9, 1 | 7,991,431 |

3, 3 | 4,442,562 | 9, 3 | 6,372,941 |

3, 7 | 7,043,695 | 9, 7 | 6,012,739 |

3, 9 | 7,502,896 | 9, 9 | 4,622,916 |

Needless to say, these figures do not match the expected uniform distribution of 6,250,000 each, with standard deviation = sqrt (10^{8} x 1/16 x 15/16) = 2420.61, approximately.

Andrew Granville of the University of Montreal expressed the common reaction of mathematicians in the field: “In ignorance, we thought things would be roughly equal. … One certainly believed that in a question like this we had a very strong understanding of what was going on.” Similarly, James Maynard of the University of Oxford, when first told of the discovery, said “I only half believed him. … As soon as I went back to my office, I ran a numerical experiment to check this myself.”

Lemke Oliver and Soundararajan believe that they now understand this phenomenon — it follows as a consequence of the “k-tuple conjecture” of 20th century British mathematicians G. H. Hardy and J. E. Littlewood, although significant extra work and analysis was required. This conjecture provides precise estimates of how often “constellations” of primes with a certain spacing will appear. Their conjecture is strongly believed to be true but has never been proven. A rigorous statement of the conjecture is presented at the MathWorld site.

This discovery highlights the importance of the experimental approach to modern mathematical research. As Lemke Oliver and Soundararajan found, well-designed computations can discover facts about the mathematical universe that are completely unexpected and counter-intuitive. And with the steadily increasing power of modern computer hardware and software, we can expect that such discoveries will only increase in the years ahead.

For additional details, see well-written articles in Quanta magazine and New Scientist. The full technical paper by Lemke Oliver and Soundararajan is available from the Arxiv site. Both authors lecture on their findings at the Alladi birthday conference, University of Florida, March 17 to 22, 2016.

[Added 16 Mar 2016:] Paul Abbott of the University of Western Australia has pointed out to us that the erratic behavior of these digits was noted in a paper by Chung-Ming Ko in 2001. Abbott notified Lemke Oliver and Soundararajan, who have added the reference to their paper.

]]>Numerous celebrations are scheduled for Pi Day 2016. San Francisco’s Exploratorium features several events, culminating with a “Pi Procession” at 1:59pm Pacific

Continue reading Pi Day 2016

]]>Once again Pi Day (March 14, or 3/14 in United States notation) is here, when both professional mathematicians and students in school celebrate this most famous of mathematical numbers. Last year was a particularly memorable Pi Day, since 3/14/15 gets two more digits correct, although some would argue that this year’s Pi Day is also memorable, since 3/14/16 is pi rounded to four digits after the decimal point (the actual value is 3.14159265358979323846…).

Numerous celebrations are scheduled for Pi Day 2016. San Francisco’s Exploratorium features several events, culminating with a “Pi Procession” at 1:59pm Pacific Time (corresponding to 3.14159) and pie served at 2:15pm. The Illinois Science Council is sponsoring a Pi Day Pi K Fun Run, starting 6:28pm (= 2 x Pi) at any of four different locations in Chicago. The website teachpi.org lists 50 ideas to make Pi Day “entertaining, educational, tasty and fun.”

Pizza Hut, in conjunction with the well-known mathematician John Conway of Princeton University, has announced that on March 14 it will release three mathematics problems on its Hut Life blog (http://blog.pizzahut.com). The first person to submit correct solutions to one of the three problems will have a chance to receive 3.14 years of free pizza from Pizza Hut. The problems vary in difficulty from high school to Ph.D. level. For details, see press report.

In past years, the present authors have celebrated Pi Day with popular articles and more serious technical pieces. For Pi Day 2014, we presented presented several examples of “piems,” i.e., poems rhapsodizing on the wonders of pi. We also presented some examples of recent research using high-tech graphical tools to explore the intriguing question of whether and why the digits of pi is “normal” (i.e., its digits are statistically “random” in a particular sense).

For Pi Day 2015, we summarized recent computations of pi, including the current world’s record of 13.3 trillion decimal digits of pi, by someone known only as “houlouonchi” and Alexander J. Yee. We also explained what pi is actually used for in modern science and technology — for example, the binary value of pi is buried in the software controlling your mobile phone, so that it can process gigahertz signals that transmit data.

Growing interest in Pi Day mirrors a growing interest in pi (and mathematics in general) in the popular culture. Where once only professional mathematicians and teachers cared about pi, nowadays it is not at all uncommon to see pi mentioned in the movies, on television or in social media. In 2003, the first three digits of pi (314) appeared in the Matrix Reloaded. On 9 May 2013, the North American quiz show Jeopardy! featured an entire category of questions on pi.

Pi has been featured in at least six different episodes of the North American TV show Person of Interest. In one episode, a secret entrance to an underground control room is guarded by what appears to be an abandoned vending machine. After entering the digits “3141,” a door opens.

In another key episode, the show’s lead character Harold Finch (played by Michael Emerson), posing as a high school teacher, explains to the class that the digits of pi are a never-ending, nonrepeating sequence, and thus all of one’s personal information, such as one’s telephone number or Social Security number appear somewhere in pi. (Note: This presumes that Pi is a “normal” number, which although widely assumed to be true, has never been proven and remains an area of active research.)

Many readers might be surprised that anything new remains to be discovered about pi. After all, given that researchers have been scouring the mathematical universe for facts about pi for millennia, surely everything of any note has already been found? This sentiment was even expressed, for example, in Petr Beckmann’s 1976 book A History of Pi. Yet almost before the ink was dry, two researchers (Gene Salamin and Richard Brent) independently discovered a new algorithm for computing pi, each iteration of which approximately *doubles* the number of correct digits. This simple algorithm can even be implemented on a programmable calculator. Numerous other, even more remarkable, formulas and algorithms for pi have been discovered in the years since 1976.

Similarly, Daniel Shanks who himself had computed pi to over 100,000 digits in 1961, once declared that computing pi to one billion digits would be “forever impossible.” Yet this was done, by both Kanada and the Chudnovsky brothers, in 1989.

Well-known mathematical physicist Roger Penrose, in the first edition of his book The Emperor’s New Mind, suggested that mankind would likely never know whether a string of ten consecutive sevens appears in the expansion of pi. But within months after the book’s publication, Kanada had indeed found the string “7777777777” in a 580-million-digit computation he had done.

Even within the past 20 years, some remarkable new facts have been discovered, some of which were completely unanticipated. One example was the 1997 discovery of a formula and associated algorithm for pi that permits one to directly calculate digits of pi starting at an arbitrary position, such as the millionth or the billionth digit, without needing to compute any of the digits that came before (although it only works with binary or base-16 digits).

With advancing computer technology, even more powerful tools are being brought to bear on pi. One example is new graphical-based techniques that permit researchers to visually explore questions such as whether and why the digits of pi are “normal” (i.e., statistically random in a certain sense, as mentioned above). We can only expect that new and heretofore unanticipated facts will come to light, perhaps sooner than many think.

To celebrate Pi Day 2016, we have decided to collect 25 key technical papers that have appeared over the past 40 years with research results related to the computation and analysis of this memorable number. The collection includes papers describing the results and computations mentioned above, and many others as well. Some papers in our collection explore advanced research questions suitable for active researchers in the field, but several are highly accessible to undergraduate mathematics students. A listing and brief description of the individual papers is available here.

]]>Here is a synopsis of the book, as taken from the Springer site:

This book contains compendium of 25 papers published since the 1970s dealing with pi and associated topics of mathematics and computer science. The collection begins with a Foreword by Bruce Berndt.

Continue reading New compendium of Pi papers

]]>Here is a synopsis of the book, as taken from the Springer site:

This book contains compendium of 25 papers published since the 1970s dealing with pi and associated topics of mathematics and computer science. The collection begins with a Foreword by Bruce Berndt. Each contribution is preceded by a brief summary of its content as well as a brief key word list indicating how the content relates to others in the collection. The collection includes articles on actual computations of pi, articles on mathematical questions related to pi (e.g., “Is pi normal?”), articles presenting new and often amazing techniques for computing digits of pi (e.g., the “BBP” algorithm for pi, which permits one to compute an arbitrary binary digit of pi without needing to compute any of the digits that came before), papers presenting important fundamental mathematical results relating to pi, and articles presenting new, high-tech techniques for analyzing pi (i.e., new graphical techniques that permit one to visually see if pi and other numbers are “normal”).

This volume is a companion to Pi: A Source Book whose third edition released in 2004. The present collection begins with 2 papers from 1976, published by Eugene Salamin and Richard Brent, which describe “quadratically convergent” algorithms for pi and other basic mathematical functions, derived from some mathematical work of Gauss. Bailey and Borwein hold that these two papers constitute the beginning of the modern era of computational mathematics. This time period (1970s) also corresponds with the introduction of high-performance computer systems (supercomputers), which since that time have increased relentlessly in power, by approximately a factor of 100,000,000, advancing roughly at the same rate as Moore’s Law of semiconductor technology. This book may be of interest to a wide range of mathematical readers; some articles cover more advanced research questions suitable for active researchers in the field, but several are highly accessible to undergraduate mathematics students.

Here is a brief description of the individual articles:

- A 1976 paper by Eugene Salamin, who exhibited an an algorithm for computing pi based on formulas with the remarkable property that each application of the formula approximately
*doubles*the number of correct digits in the result. - A 1976 paper by Richard Brent, who presented his independently discovered version of the algorithm described above, plus similar “quadratically convergent” algorithms for computing other transcendental functions and constants.
- A 1984 paper by David Cox, who gives some interesting background on Gauss’ discovery of the arithmetic-geometric mean and its connection to the theory of elliptic modular functions.
- A 1984 paper by one of us (Jonathan Borwein) and his brother (Peter Borwein), presenting some even faster algorithms for computing Pi and various elementary functions such as sine, cosine, exponential and logarithm. This paper was read by the other of us (Bailey), which led to some large-scale computations of Pi and a 32-year-and-counting collaboration.
- A 1985 paper by D. J. Newman, who presented simplified proofs of the algorithms of Salamin and Brent.
- A 1985 paper by Stan Wagon of Macalester College in Minnesota, who highlighted the long-standing and unresolved question of whether (and why) the digits of Pi are statistically random in a certain specific sense.
- A 1988 paper by one of us (Bailey) describing the computation of Pi to 29,360,000 digits, which at the time was a world’s record. This computer program was used as part of an integrity test of a new supercomputer and, in fact, did disclose some hardware bugs in the system.
- A paper by Gert Almkvist and Bruce Berndt on the historical roots of Gauss’ arithmetic-geometric mean, which is the basis of several state-of-the-art methods for computing Pi. Among these roots was the writings of a 19th century British mathematician who published articles for the
*Ladies Diary*, a popular womens magazine that featured a regular mathematics column. - A 1988 article by Japanese computer scientist Yasumasa Kanada, who computed Pi to 201,326,000 digits (a record at the time).
- A 1988
*Scientific American*article describing how the work of Indian mathematician Srinivasa Ramanujan has been applied to the computation of Pi. - A 1989 paper by the present authors and Peter Borwein from the
*American Mathematical Monthly*that presents the historical and mathematical background of Pi, including a formula discovered by Ramanujan that was used, after suitable modification, to compute up to two billion digits of Pi. - A 1989 paper that highlighted and explained the remarkable fact that when Gregory formula for Pi, a simple formula dating back to 1671, is used to compute decimal digits of Pi using, say, 500,000 terms, the errors exhibit a curious pattern, with correct digits interspersed by errors.
- A 1995 paper by Stanley Rabinowitz and Stan Wagon that exhibited a simple “spigot algorithm” for the digits of pi that generates the digits one by one, much like drops of water dripping from a water spigot.
- A 1997 paper by Bailey, Peter Borwein and Simon Plouffe that presented what is now known as the “BBP formula” for Pi. This formula has the remarkable property that it permits one to calculate binary or base-16 digits of Pi beginning at an arbitrary starting position (such as the millionth or the billionth position), without needing to compute any of the digits that came before. Included in this paper were base-16 digits of Pi beginning at position ten billion, a record at the time.
- A 2001 paper by Dirk Huylebrouck that presents, in a relatively simple, self-contained fashion, proofs of irrationality (i.e., proofs that the constant cannot be written as the ratio of two whole numbers) for Pi, the natural logarithm of 2, and the Riemann zeta function evaluated at 2 and 3.
- 2006 paper by Jeremy Gibbons, who found “unbounded” variations of of the Rabinowitz-Wagon “spigot” algorithm.
- Two selections from a 2008 book by the present authors that deal with Pi: one chapter deals specifically with formulas (with proofs) for computing Pi; the other deals with the persistent and still-unresolved question of whether Pi and various other mathematical constants are “normal” (i.e., have statistically random digits in a certain sense).
- A 2009 paper by Stephen Lucas, who showed how approximations for Pi such as 22/7 and 355/113 can be seen to follow straightforwardly from integral formulas.
- A 2009 paper by Baruah, Berndt and Chan discussing a formula of Ramanujan for Pi and a related formula that was employed by David and Gregory Chudnovsky to compute over two billion decimal digits of Pi.
- A 2013 paper by the present authors, Andrew Mattingly and Glenn Wightwick describing the computation of base-64 digits of Pi
^{2}, base-729 digits of Pi^{2}and base-4096 digits of Catalan’s constant, in each case beginning with position*ten trillion*. - A 2013 paper by Francisco Aragon Artacho, the present authors and Peter Borwein analyzing the normality (a form of statistical randomness) of the digits of Pi and numerous other mathematical constants using new techniques based on computer graphics.
- A 2013 article by Ravi Agarwal, Haans Agarwal and Syamal K. Sen presenting a detailed chronology of the computation and analysis of Pi from its earliest origins in India and Babylonian mathematics up to and in eluding the most recent results and computations.
- A 2014 article by us from the
*American Mathematical Monthly*giving a brief overview of the history of Pi and decimal arithmetic, up to and including the high-tech methods, such as graphical analysis, that are now being employed by researchers investigating the normality of Pi. - A 2014 by Jonathan Borwein, entitled “The Life of Pi,” which presents numerous interesting and important facts about Pi in their historical context.
- A 2014 paper by Jonathan Borwein and Scott Chapman, entitled “I prefer Pi” (a palindrome), listing and briefly describing the many papers on Pi that have appeared in the
*American Mathematical Monthly*, the world’s most widely read mathematics journal.