Prime numbers are the very atoms of arithmetic. Their importance to mathematics comes from their power to build all other numbers. Mastering these building blocks offers the mathematician the hope of discovering new ways of charting a course through the vast complexities of the mathematical world. Yet prime numbers remain the most mysterious objects studied by mathematicians. It is impossible for one looking through a list of prime numbers to predict when the next prime will appear. The list seems chaotic, random, and offers no clues as to how to determine the next number.
The list of primes is the heartbeat of mathematics, but its pulse is irregular. In Carl Sagan’s classic novel Contact, aliens use prime numbers to contact intelligent life on Earth by repeatedly beaming a radio signal through the cosmos with a sequence of prime numbers up to 907 to attract the attention of earthlings. Indeed, generations of mathematicians have sat listening to the rhythm of the ‘prime number drum’ as it beats out its sequence of numbers: two, three, five, seven, eleven, … Even after millennia of intense mathematical work, no one has been able to identify a pattern among these numbers.
A tantalizing insight into the pattern was, however, produced almost 150 years ago by the great German mathematician Bernhard Riemann when he published a remarkable hypothesis, which – if true – says that the primes have music in them. Marcus du Sautoy in his delightful book "The Music of the Primes" describes a special three-dimensional landscape, i.e., Riemann’s treasure map of the primes, where they all seem to be miraculously arranged in a straight line as far as the eyes could see, and each point at sea level correspond to a musical note. It is the combination of these notes, each at just the right volume, that give rise to the music of the primes. If Riemann's landscape were true, the orchestra playing the music of the primes will be in perfect balance. It is as if each instrument plays its own pattern, but by combining together so perfectly, the patterns cancel themselves out, leaving just the formless ebb and flow of the primes.
Riemann looked at the image of the primes in the mirror that separated the world of numbers from his imaginary landscape, and saw the seemingly random arrangement of prime numbers on one side of the mirror transform into the strict regimented order of the zeroes on the other side of the mirror. Riemann could not have anticipated what was awaiting him on the other side of the looking glass. But what lay there completely transformed the task of understanding the mysteries of the primes. And mathematicians now had a new landscape to explore.
- du Sautoy, Marcus (2012). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Perennial.
- Hofstadter, Douglas R. (1979). Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books.