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Friday, April 06, 2018

Why Prime Numbers Still Surprise and Mystify Mathematicians | Smithsonian

This article was originally published on The Conversation.


Photo: Martin H Weissman
Martin H. Weissman, Associate Professor of Mathematics, University of California, Santa Cruz inform, "2300 years later, new patterns continue to show up in these indivisible tricksters"

Primes still have the power to surprise.
Photo: Chris-LiveLoveClick/shutterstock.com

On March 20, American-Canadian mathematician Robert Langlands received the Abel Prize, celebrating lifetime achievement in mathematics. Langlands’ research demonstrated how concepts from geometry, algebra and analysis could be brought together by a common link to prime numbers.

Photo: Robert Langlands addresses a conference at the Institute for Advanced Study in 2016.

When the King of Norway presents the award to Langlands in May, he will honor the latest in a 2,300-year effort to understand prime numbers, arguably the biggest and oldest data set in mathematics. As a mathematician devoted to this “Langlands program, I’m fascinated by the history of prime numbers and how recent advances tease out their secrets. Why they have captivated mathematicians for millennia?

To study primes, mathematicians strain whole numbers through one virtual mesh after another until only primes remain. This sieving process produced tables of millions of primes in the 1800s. It allows today’s computers to find billions of primes in less than a second. But the core idea of the sieve has not changed in over 2,000 years.

“A prime number is that which is measured by the unit alone,” mathematician Euclid wrote in 300 B.C. This means that prime numbers can’t be evenly divided by any smaller number except 1. By convention, mathematicians don’t count 1 itself as a prime number. Euclid proved the infinitude of primesthey go on foreverbut history suggests it was Eratosthenes who gave us the sieve to quickly list the primes.

Here’s the idea of the sieve. First, filter out multiples of 2, then 3, then 5, then 7—the first four primes. If you do this with all numbers from 2 to 100, only prime numbers will remain.

With eight filtering steps, one can isolate the primes up to 400. With 168 filtering steps, one can isolate the primes up to 1 million. That’s the power of the sieve of Eratosthenes.

An early figure in tabulating primes is John Pell, an English mathematician who dedicated himself to creating tables of useful numbers. He was motivated to solve ancient arithmetic problems of Diophantos, but also by a personal quest to organize mathematical truths. Thanks to his efforts, the primes up to 100,000 were widely circulated by the early 1700s. By 1800, independent projects had tabulated the primes up to 1 million.

To automate the tedious sieving steps, a German mathematician named Carl Friedrich Hindenburg used adjustable sliders to stamp out multiples across a whole page of a table at once. Another low-tech but effective approach used stencils to locate the multiples. By the mid-1800s, mathematician Jakob Kulik had embarked on an ambitious project to find all the primes up to 100 million.

This “big data” of the 1800s might have only served as reference table, if Carl Friedrich Gauss hadn’t decided to analyze the primes for their own sake. Armed with a list of primes up to 3 million, Gauss began counting them, one “chiliad,” or group of 1,000 units, at a time. He counted the primes up to 1,000, then the primes between 1,000 and 2,000, then between 2,000 and 3,000 and so on.
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Additional resources 
Why prime numbers still fascinate mathematicians, 2,300 years later | Justin Trudeau Stories 

Source: Smithsonian and Justin Trudeau Stories Channel (YouTube)


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