(Roughly) Daily

Posts Tagged ‘Number theory

“I have had my results for a long time, but I do not yet know how to arrive at them”*…



Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into the third talk, he came to the end. Then, to punctuate the result, he added:

=> FLT

“Implies Fermat’s Last Theorem.” The most famous unverified conjecture in the history of mathematics. First proposed by the 17th-century French jurist and spare-time mathematician Pierre de Fermat, it had remained unproven for more than 350 years. Wiles, a professor at Princeton University, had worked on the problem, alone and in secret in the attic of his home, for seven years. Now he was unveiling his proof.

His announcement electrified his audience—and the world. The story appeared the next day on the front page of The New York Times. Gap, the clothing retailer, asked him to model a new line of jeans, though he demurred. People Weekly named him one of “The 25 Most Intriguing People of the Year,” along with Princess Diana, Michael Jackson, and Bill Clinton. Barbara Walters’ producers reached out to him for an interview, to which Wiles responded, “Who’s Barbara Walters?”

But the celebration didn’t last. Once a proof is proposed, it must be checked and verified before it is accepted as valid. When Wiles submitted his 200-page proof to the prestigious journal Inventiones Mathematicae, its editor divvied up the manuscript among six reviewers. One of them was Nick Katz, a fellow Princeton mathematician.

For two months, Katz and a French colleague, Luc Illusie, scrutinized every logical step in Katz’s section of the proof. From time to time, they would come across a line of reasoning they couldn’t follow. Katz would email Wiles, who would provide a fix. But in late August, Wiles offered an explanation that didn’t satisfy the two reviewers. And when Wiles took a closer look, he saw that Katz had found a crack in the mathematical scaffolding. At first, a repair seemed straightforward. But as Wiles picked at the crack, pieces of the structure began falling away…

How mistakes– first Fermat’s, then Wiles’– reinvigorated a field, then led to fundamental insight: “How Math’s Most Famous Proof Nearly Broke.”

* Karl Friedrich Gauss


As we ponder proof, we might we might spare a thought for Josiah Wedgwood; he died on this date in 1795. An English potter and businessman (he founded the Wedgwood company), he is credited, via his technique of “division of labor,” with the industrialization of the manufacture of pottery– and via his example, much of British (and thus American) manufacturing.

Wedgwood was a member of the Lunar Society, the Royal Society, and was an ardent abolitionist.  His daughter, Susannah, was the mother of Charles Darwin.



“Exploring pi is like exploring the universe”*…




Pi is an infinite string of seemingly random numbers, but if you break down the first 1000 digits of Pi according to how many times each number from 0 to 9 appears, they’re all just about equal — with 1 being the outlier at 12% (although we wonder if they’d all average to ~10% given enough digits of Pi)…

More at “Visualizing The Breakdown Of The Numbers In The First 1000 Digits Of Pi Is Fascinating.”

* David Chudnovsky


As we watch it even out in the end, we might spare a thought for Hannah Wilkinson Slater; she died on this date in 1812. The daughter and the wife of mill owners, Ms. Slater was the first woman to be issued a patent in the United States (1793)– for a process using spinning wheels to twist fine Surinam cotton yarn, that created a No. 20 two-ply thread that was an improvement on the linen thread previously in use for sewing cloth.

A waxen Hannah, at the Slaters’ Mill Museum in Pawtucket, RI




Written by LW

October 2, 2017 at 1:01 am

“Mathematics is the queen of sciences and number theory is the queen of mathematics”*…


The Online Encyclopedia of Integer Sequences.  Because.

(Visit the page of its parent, The OEIS Foundationmovies, posters, and more!)

* Carl Friedrich Gauss


As we count ’em up, we might send starry birthday greeting to Erasmus Reinhold; he was born on this date in 1511.  A mathematician and astronomer, Reinhold was considered to be the most influential astronomical pedagogue of his generation. Today, he is probably best known for his carefully calculated set of planetary tables– the first– applying Copernican theory, published in 1551.



“Prime numbers are what is left when you have taken all the patterns away”*…


Der Hl. Augustinus und der Teufel


1, followed by 13 zeros, then 666, and then another 13 zeros, and a final 1:  a palindromic prime number named for Belphegor (or Beelphegor), one of the seven princes of Hell.  Reputed to help people make discoveries, Belphegor is the demon of inventiveness.  He figures in Milton’s Paradise Lost as the namesake of one of the “Principalities of the Prime”… So it is only fitting that these devilish digits bear his name.

The symbol of Belphegor’s Prime resembles Pi, only upside down. It is derived from a bird glyph first seen embedded in the Voynich Manuscript.


More prime provocation at Cliff Pickover‘s “Belphegor’s Prime: 1000000000000066600000000000001.”

* Mark Haddon, The Curious Incident of the Dog in the Night-Time


As we try to divine divisors, we might recall that it was on this date in 1968 that the first-ever 9-1-1 call was placed by Alabama Speaker of the House Rankin Fite, from Haleyville City Hall, to U.S. Rep. Tom Bevill, at the city’s police station.

Emergency numbers date back to 1937, when the British began to use 999.  But experience showed that three repeated digits led to many mistaken/false alarms.  The Southern California Telephone Co. experimented in 1946 in Los Angeles with 116 for emergencies.

But 911– using just the first and last digits available– yielded the best results, and went into widespread use in the 1980s when 911 was adopted as the standard emergency number across most of the country under the North American Numbering Plan.

And yes, “911” is a prime…


Written by LW

February 16, 2015 at 1:01 am

No reservation? No problem!…


Jeff Dekofsky explains Hilbert’s paradox of the Grand Hotel, a thought experiment proposed in the 1920s by German mathematician David Hilbert to illustrate some surprising properties of infinite sets, in this TED-Ed animated lecture

email readers click here for video

As a special bonus, another amusing video (via Kottke)– an explanation of why it is that the sum of all positive integers (1 + 2 + 3 + 4 + 5 + …) = -1/12…  Euler actually proved this result in 1735, but the result was only made rigorous later; and now physicists have been seeing this result actually show up in nature.  (Spoiler alert: the answer turns on what one means by “sum” mathematically…)


As we pray for more fingers and toes, we might spare a thought for Harald August Bohr; he died on this date in 1951.  While materially less well-known than his brother Niels, Harald was a formidable mathematician (founder of the field of almost periodic functions), a gifted athlete (an accomplished footballer who won a silver medal at the 1908 Summer Olympics as a member of Denmark’s team), an inspirational teacher (the annual award for outstanding teaching at the University of Copenhagen is called “the Harald” in his honor), and an out-spoken critic of the anti-Semitic policies that took root in the German mathematical establishment in the 1930s.


What’s (the) matter?…


On the heels of yesterday’s film recommendation, another… albeit somewhat different:  Stanford physics professor, Leonard Susskind, one of the fathers of string theory, articulator of the Holographic Principle,  and explainer of the Megaverse, has a gift for making science accessible… a gift that is on display in this lecture, “Demystifying the Higgs Boson“:

(email readers, click here)


As we say “ahh,” we might spare a thought for Pierre de Fermat; he died on this date in 1665.  With Descartes, one of the two great mathematicians of the first half of the Seventeenth Century, Fermat made a wide range of contributions (that advanced, among other fronts, the development of Calculus) and is regarded as the Father of Number Theory.  But he is best remembered as the author of Fermat’s Last Theorem.* Fermat had written the theorem, in 1637, in the margin of a copy of Diophantus’ Arithmetica– but went on to say that, while he had a proof, it was too large to fit in the margin.  He never got around to committing his proof to writing; so mathematicians started, from the time of his death, to try to derive one.  While the the theorem was demonstrated for a small number of cases early on, a complete proof became the “white whale” of math, eluding its pursuers until 1995, when Andrew Wiles finally published a proof.

* the assertion that no three positive integers ab, and c can satisfy the equation an + bn = cn for any integer value of n greater than two


Written by LW

January 12, 2013 at 1:01 am

Pieces of pi…

In 2010, Japanese engineer Shigeru Kondo set a record, calculating the value of pi to 5 trillion digits… then last October, he smashed his own mark, identifying the first 10 trillion decimal places.  (He used a home-made computer that ran so hot that the temperature in his apartment was over 100 degrees…)

The quest will no doubt continue– pi is an irrational number that exerts an irrational fascination.  Meantime, readers can take a peek at this work-perpetually-in-progress.  Web design firm firm Two-N has created this nifty visualization and search tool, allowing one to find any one of the first 4,000,000 digits of pi:

Bonus: “50 Interesting Facts About Pi


As we ruminate on randomness, we might send carefully-calculated birthday greetings to Hermann Minkowski; he was born on this date in 1864.  Minkowski developed the geometry of numbers and used geometrical methods to solve difficult problems in number theory and mathematical physics; he is probaly best remembered for realizing that his former student Albert Einstein’s special theory of relativity (1905), presented algebraically by Einstein, could also be understood geometrically as a theory of four-dimensional space-time.  Einstein embraced the geometric approach in the development of his theory of general relativity– and the four-dimensional space (the three physical dimensions plus time) involved has since been known as “Minkowski spacetime.”

Minkowski’s best friend was “mathematical hotelier” David Hilbert.


Written by LW

June 22, 2012 at 1:01 am

%d bloggers like this: