Tag: number theory

👓 Kaisa Matomäki Dreams of Primes | Quanta Magazine

Read Kaisa Matomäki Dreams of Primes by Kevin Hartnett (Quanta Magazine)
Kaisa Matomäki has proved that properties of prime numbers over long intervals hold over short intervals as well. The techniques she uses have transformed the study of these elusive numbers.
Syndicated copies to:

Primes as a Service on Twitter

Our friend Andrew Eckford has spent some time over the holiday improving his Twitter bot Primes as a Service. He launched it in late Spring of 2016, but has added some new functionality over the holidays. It can be relatively handy if you need a quick answer during a class, taking an exam(?!), to settle a bet at a mathematics tea, while livetweeting a conference, or are hacking into your favorite cryptosystems.

General Instructions

Tweet a positive 9-digit (or smaller) integer at @PrimesAsAService. It will reply via Twitter to tell you if the number prime or not.

Some of the usable commands one can tweet to the bot for answers follow. (Hint: Click on the buttons with the tweet text to auto-generate the relevant Tweet.)

If you ask about a prime number with a twin prime, it should provide the twin.

Pro tip: You should be able to drag and drop any of the buttons above to your bookmark bar for easy access/use in the future.

Happy prime tweeting!

Syndicated copies to:

The L-functions and modular forms database

Bookmarked The L-functions and modular forms database by Tim Gowers (Gowers's Weblog)
...I rejoice that a major new database was launched today. It’s not in my area, so I won’t be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.

…I rejoice that a major new database was launched today. It’s not in my area, so I won’t be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.

–Tim Gowers

Syndicated copies to:

Devastating News: Sol Golomb has apparently passed away on Sunday

The world has certainly lost one of its greatest thinkers, and many of us have lost a dear friend, colleague, and mentor.

I was getting concerned that I hadn’t heard back from Sol for a while, particularly after emailing him late last week, and then I ran across this notice through ITSOC & the IEEE:

Solomon W. Golomb (May 30, 1932 – May 1, 2016)

Shannon Award winner and long-time ITSOC member Solomon W. Golomb passed away on May 1, 2016.
Solomon W. Golomb was the Andrew Viterbi Chair in Electrical Engineering at the University of Southern California (USC) and was at USC since 1963, rising to the rank of University and Distinguished Professor. He was a member of the National Academies of Engineering and Science, and was awarded the National Medal of Science, the Shannon Award, the Hamming Medal, and numerous other accolades. As USC Dean Yiannis C. Yortsos wrote, “With unparalleled scholarly contributions and distinction to the field of engineering and mathematics, Sol’s impact has been extraordinary, transformative and impossible to measure. His academic and scholarly work on the theory of communications built the pillars upon which our modern technological life rests.”

In addition to his many contributions to coding and information theory, Professor Golomb was one of the great innovators in recreational mathematics, contributing many articles to Scientific American and other publications. More recent Information Theory Society members may be most familiar with his mathematics puzzles that appeared in the Society Newsletter, which will publish a full remembrance later.

A quick search a moment later revealed this sad confirmation along with some great photos from an award Sol received just a week ago:

As is common in academia, I’m sure it will take a few days for the news to drip out, but the world has certainly lost one of its greatest thinkers, and many of us have lost a dear friend, colleague, and mentor.

I’ll try touch base with his family and pass along what information sniff I can. I’ll post forthcoming obituaries as I see them, and will surely post some additional thoughts and reminiscences of my own in the coming days.

Golomb and national medal of science
President Barack Obama presents Solomon Golomb with the National Medal of Science at an awards ceremony held at the White House in 2013.
Syndicated copies to:

What An Actual Handwaving Argument in Mathematics Looks Like

I’m sure we’ve all heard them many times, but this is what an actual handwaving argument looks like in a mathematical setting.

Handwaving during Algebraic Number Theory

Instagram filter used: Normal

Photo taken at: UCLA Math Sciences Building

Handwaving during Algebraic Number Theory

A photo posted by Chris Aldrich (@chrisaldrich) on

Syndicated copies to:

Shinichi Mochizuki and the impenetrable proof of the abc conjecture

Read The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof of the ABC Conjecture (Nature News & Comment)
A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right.

The biggest mystery in mathematics

This article in Nature is just wonderful. Everyone will find it interesting, but those in the Algebraic Number Theory class this fall will be particularly interested in the topic – by the way, it’s not too late to join the class. After spending some time over the summer looking at Category Theory, I’m tempted to tackle Mochizuki’s proof as I’m intrigued at new methods in mathematical thinking (and explaining.)

The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.

Thanks to Rama for bringing this to my attention!

Syndicated copies to: