How many primes can you name? In March this year mathematician Robert Langlands won the Abel Prize for research showing how concepts from different branches of mathematics all share links to prime numbers. To analyse these numbers mathematicians have to sift through numbers using mathematical filters, eliminating all non-primes. This search has its origins in antiquity; Euclid wrote in 300 BC that “a prime number is that which is measured by the unit alone”, and it was he who proved that the number of primes is infinite. However, it was probably Eratosthenes who first developed the sifting process, which filters out all multiples of 2, 3, 5, and 7—the first 4 primes.
A notable figure in the early history of the study of primes is John Pell, whose urge to categorise and collect useful numbers led him to identify and publish the primes up to 100,000 in the early 1700s. A century later, others had found the primes up to 1 million. As more and more primes were found, the process was made easier by the invention of sliders and stencils to quickly eliminate multiples. However, it was Carl Friedrich Gauss who decided to actually analyse prime numbers, looking for interesting patterns. He found, for example, that the higher he counted, the fewer prime numbers there were. More recently it has been found that, with the exception of 2 and 5, all prime numbers end in 1,3,7, or 9.
Langlands’ research, which has been described as “revolutionary”, is founded on the work of previous mathematicians, in particular Gauss. In the late 18th century he formulated a law of reciprocity whereby certain types of primes share defining characteristics; for example, primes that are the sum of two squares also leave a remainder of 1 when divided by 4. Langlands built on this by proposing that prime numbers encoded in higher-degree equations than simply squares might be in a reciprocal relationship with the branch of mathematics known as harmonic analysis, which is often used in physics.
Applicants for Mathematics may wish to read Langland’s research and look into the contemporary questions in the study of prime numbers. Students wishing to study Physics could familiarise themselves with harmonic analysis and learn about how prime numbers are relevant to physics.