{"id":108936,"date":"2018-04-16T12:00:30","date_gmt":"2018-04-16T11:00:30","guid":{"rendered":"https:\/\/www.transcend.org\/tms\/?p=108936"},"modified":"2018-04-09T12:00:13","modified_gmt":"2018-04-09T11:00:13","slug":"why-prime-numbers-still-fascinate-mathematicians-2300-years-later","status":"publish","type":"post","link":"https:\/\/www.transcend.org\/tms\/2018\/04\/why-prime-numbers-still-fascinate-mathematicians-2300-years-later\/","title":{"rendered":"Why Prime Numbers Still Fascinate Mathematicians 2,300 Years Later"},"content":{"rendered":"
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Primes still have the power to surprise.
Chris-LiveLoveClick\/shutterstock.com<\/p><\/div>\n

2 Apr 2018 – <\/em>On March 20, American-Canadian mathematician Robert Langlands received the Abel Prize<\/a>, celebrating lifetime achievement in mathematics. Langlands\u2019 research demonstrated how concepts from geometry, algebra and analysis could be brought together by a common link to prime numbers.<\/p>\n

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.<\/p>\n

As a mathematician devoted to this \u201cLanglands program,\u201d<\/a> I\u2019m fascinated by the history of prime numbers and how recent advances tease out their secrets. Why they have captivated mathematicians for millennia?<\/p>\n

How to find primes<\/strong><\/p>\n

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\u2019s computers to find billions of primes in less than a second<\/a>. But the core idea of the sieve has not changed in over 2,000 years.<\/p>\n

\u201cA prime number is that which is measured by the unit alone,\u201d mathematician Euclid wrote in 300 B.C.<\/a> This means that prime numbers can\u2019t be evenly divided by any smaller number except 1. By convention, mathematicians don\u2019t count 1 itself as a prime number.<\/p>\n

Euclid proved the infinitude of primes \u2013 they go on forever \u2013 but history suggests it was Eratosthenes<\/a> who gave us the sieve to quickly list the primes.<\/p>\n

Here\u2019s the idea of the sieve. First, filter out multiples of 2, then 3, then 5, then 7 \u2013 the first four primes. If you do this with all numbers from 2 to 100, only prime numbers will remain.<\/p>\n

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Sieving multiples of 2, 3, 5 and 7 leaves only the primes between 1 and 100. Courtesy of M.H. Weissman.<\/p><\/div>\n

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\u2019s the power of the sieve of Eratosthenes.<\/p>\n

Tables and tables<\/strong><\/p>\n

An early figure in tabulating primes<\/a> 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.<\/p>\n

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<\/a> had embarked on an ambitious project to find all the primes up to 100 million.<\/p>\n

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A stencil used by Kulik to sieve the multiples of 37. AO\u0308AW, Nachlass Kulik
Image courtesy of Denis Roegel, Author provided<\/p><\/div>\n

This \u201cbig data\u201d of the 1800s might have only served as reference table, if Carl Friedrich Gauss<\/a> hadn\u2019t decided to analyze the primes for their own sake. Armed with a list of primes up to 3 million, Gauss began counting them, one \u201cchiliad<\/a>,\u201d or group of 1000 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.<\/p>\n

Gauss discovered that, as he counted higher, the primes gradually become less frequent according to an \u201cinverse-log\u201d law. Gauss\u2019s law doesn\u2019t show exactly how many primes there are, but it gives a pretty good estimate. For example, his law predicts 72 primes between 1,000,000 and 1,001,000. The correct count is 75 primes<\/a>, about a 4 percent error.<\/p>\n

A century after Gauss\u2019 first explorations, his law was proved in the \u201cprime number theorem.\u201d<\/a> The percent error approaches zero at bigger and bigger ranges of primes. The Riemann hypothesis, a million-dollar prize problem<\/a> today, also describes how accurate Gauss\u2019 estimate really is.<\/p>\n

The prime number theorem and Riemann hypothesis get the attention and the money, but both followed up on earlier, less glamorous data analysis.<\/p>\n

Modern prime mysteries<\/strong><\/p>\n

Today, our data sets come from computer programs rather than hand-cut stencils, but mathematicians are still finding new patterns in primes.<\/p>\n

Except for 2 and 5, all prime numbers end in the digit 1, 3, 7 or 9. In the 1800s, it was proven that these possible last digits are equally frequent. In other words, if you look at the primes up to a million, about 25 percent end in 1, 25 percent end in 3, 25 percent end in 7, and 25 percent end in 9.<\/p>\n

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