A Tale of the Primes — Part 3

Previous

Riemann died seven years after he introduced the Riemann Hypothesis, leaving the world without a proof. And so it was, as the 19th century gradually came to a close. In 1900, David Hilbert included the Riemann hypothesis in his famous list of 23 unsolved problems. The list was intended to shape the direction of mathematics for the new century. The Riemann Hypothesis is part of Problem 8 in the 23 problems. Hilbert said of the problem: “If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann Hypothesis been proven?”

Hilbert was not exaggerating. The Riemann Hypothesis resisted all attacks in the 20th century. It even threw some contestants into madness. According to Sylvia Nasar’s biography, A Beautiful Mind, mathematician John Forbes Nash was captivated by the prospect of proving the hypothesis and soon became delusional of having obtained a solution. Nash presented his solution to the American Mathematical Society in 1959, stunning fellow mathematicians with a series of nonsensical sentences. Less dramatic yet equally frustrating stories were all too common.

At the conclusion of the 20th century, the Riemann Hypothesis remained the only unsolved problem on Hilbert’s list. In 2000, the Clay Mathematics Institute of Cambridge, Massachusetts announced the Riemann Hypothesis as part of a new list of 7 problems for mathematicians in the new millennium. Attached to each problem is a $1,000,000 prize. Now, a proof of the Riemann Hypothesis is not only worth satisfaction and recognition, but also one million dollars.

But there is far more at stake. In the 1970s, the study of prime numbers was suddenly pulled out of the ivory towers of pure mathematics and into the rough and dirty world of commerce. Three MIT scientists exploited one of Fermat’s many discoveries about prime numbers and constructed a code that would provide impeccable security for sensitive information traveling over the Internet. This cryptography method is now commonly adopted by e-businesses and national intelligence agencies. The method relies on the fact that our knowledge of the primes is just enough to construct the code, but too little to crack it. Since the Riemann Hypothesis provides significant insight to the prime numbers, it is now the center of attention for the likes of AT&T, Hewlett-Packard, and the US National Security Agency.

As for the everyday online shopper, there is little need to worry. The proving of the Riemann Hypothesis has no immediate impact on Internet security. By the time its long term effects trickle down to the cracking of Internet security codes, governments and businesses will have had plenty of time to replace all cryptography methods involving the prime numbers. (The mathematical objects known as elliptical curves, for example, are ready alternatives for the prime numbers.) So, as the study of the primes continues, we may continue to enjoy the convenience and security of online shopping.

Previous