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Posts Tagged ‘Number theory

“He told me that in 1886 he had invented an original system of numbering”*…

A visualization of the 3-adic numbers

The rational numbers are the most familiar numbers: 1, -5, ½, and every other value that can be written as a ratio of positive or negative whole numbers. But they can still be hard to work with.

The problem is they contain holes. If you zoom in on a sequence of rational numbers, you might approach a number that itself is not rational. This short-circuits a lot of basic mathematical tools, like most of calculus.

Mathematicians usually solve this problem by arranging the rationals in a line and filling the gaps with irrational numbers to create a complete number system that we call the real numbers.

But there are other ways of organizing the rationals and filling the gaps: the p-adic numbers. They are an infinite collection of alternative number systems, each associated with a unique prime number: the 2-adics, 3-adics, 5-adics and so on.

The p-adics can seem deeply alien. In the 3-adics, for instance, 82 is much closer to 1 than to 81. But the strangeness is largely superficial: At a structural level, the p-adics follow all the rules mathematicians want in a well-behaved number system…

“We’re all on Earth and we work with the reals, but if you went [anywhere] else, you’d work with the p-adics,” [University of Washington mathematician Bianca] Viray explained. “It’s the reals that are the outliers.”

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory… which is itself at the heart of computer science, numerical analysis, and cryptography: “An Infinite Universe of Number Systems.”

* Jorge Luis Borges, Labyrinths


As we dwell on digits, we might send carefully-calculated birthday greetings to Klaus Friedrich Roth; he was born on this date in 1925. After escaping with his family from Nazi Germany, he was educated at Cambridge, then taught mathematics first at University College London, then at Imperial College London. He made a number of important contribution to Number Theory, for which he won the De Morgan Medal and the Sylvester Medal, and election to Fellowship of the Royal Society. In 1958 he was awarded mathematics’ highest honor, the Fields Medal, for proving Roth’s theorem on the Diophantine approximation of algebraic numbers.


“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

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