Prime numbers––those divisible only by themselves and one––have confounded mathematicians for centuries. Because mathematicians rely on patterns, the fact that primes occur at seemingly random intervals (2, 3, 5, 7, 11, 13…) makes them the Holy Grail of math. Many who studied the numbers burned out in their primes, fell into depressions, or attempted suicide.
The Music of the Primes, a companion to the BBC documentary The Story of Math, delves into the history of mathematicians’ struggle to understand the primes. In 300 B.C., Greek mathematician Euclid proved that there are an infinite number of primes, but could not find a way to predict when they would show up in a sequence.
At the beginning of the 19th century, German scientist Carl Friedrich Gauss made a major breakthrough: instead of asking which numbers are prime, he asked how many are. Using a graph like the one below, Gauss predicted the rate at which primes thin out as numbers become larger. Program host Marcus du Sautoy tells us that Gauss “heard the dominant theme of the music of the primes, but he couldn’t prove it.” But we must wait to understand this the meaning of this statement.
About fifty years later, mathematician Bernhard Riemann took Gauss’ predictions one step further. Using the physics of waves of musical tones as a guide, Riemann came up with a way to give order to the distribution of the primes. For an explanation of the hypothesis, see the clip below from the documentary and read this article from Marcus du Sautoy.
Riemann’s Hypothesis was a revelation for mathematicians, but it remained just that: a hypothesis. He couldn’t prove it. Nevertheless, his parade of zeroes was a major revelation: Reimann had connected two disparate realms of mathematics––zeros and primes. When Reimann died of tuberculosis at 39, his housekeeper burned all his papers, so we’ll never know how close he was to a proof. His hypothesis became one of the greatest unsolved mathematical mysteries.
In the 1940s, British mathematician and computer pioneer Alan Turing approached the problem in a new way. He tried to prove Reimann’s Hypothesis false by building a machine that would search for rogue zeros off the line. After World War II, Turing’s machine showed that the first 1,104 zeros were on the line, but then the machine broke down. So did Turing’s life; he was persecuted for his homosexuality and ultimately committed suicide.
A major breakthrough occurred in the 1970s at Princeton. Driving a red sports car, blaring the punk rock song “American Idiot,” Du Sautoy arrives at the university and interviews Hugh Montgomery, who had noticed that the scattered zeroes on the line seemed to repel one another. He consulted physicist Freeman Dyson, who recognized that the pattern was strangely similar to a matrix used to model the nucleus of Uranium.
Here, the meaning of music of the primes finally begins to emerge. The energy levels of the nucleus of an atom are like musical notes, Du Sautoy says, and then plays several on his trumpet to illustrate. “As I blow more energy into it, the notes jump up by degrees,” he says. The energy levels of the nucleus space out, just as zeros on the line do, he explains. “The behavior of the fundamental building blocks of matter seemed to correspond to the fundamental behavior of the building blocks of maths.” This is the crescendo of the documentary (pun intended).
Despite this contribution, the riddle of the primes persists, and today mathematicians continue to devote vast amounts of time and computational power to the Riemann Hypothesis. Most believe it to be true, but no one has yet found a proof. A financier has offered $1 million to anyone who can crack the hypothesis. Du Sautoy believes that who ever does will make the primes sing.
The Music of the Primes is available on DVD here.