What is the best number?

“The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3… and in binary 73 is a palindrome, 1001001, which backwards is 1001001.”

Sheldon Cooper, The Big Bang Theory



That prime numbers are funny objects is no recent discovery. Some of their basic properties, such as the fact that there is an infinite number of primes, could be proven with relative ease by Euclid of Alexandria already around 300 BC.

(How much ease? Check out the proof yourself!)

Other empirical facts had to wait two thousand years before a rigorous proof could be found; others yet have survived till our days only as unproved conjectures. Here’s a small sample:

  • Between any number and its double, there is always at least one prime number (Bertrand’s conjecture, proved by Chebishev in 1850);
  • Every even integer greater than two can be expressed as the sum of two primes (Goldbach’s conjecture, yet unproven);
  • Prime numbers can occur as twins (i.e., separated by 2, like 3 and 5), cousins (separated by 4), and so on. Whether these subsets are themselves infinite is also still a conjecture;
  • The number of prime numbers smaller than a certain number n is approximated by n/log(n) as n grows (Prime number theorem, proved by Hadamard and de la Vallée-Poussin in 1896);

A recent breakthrough on the distribution of prime numbers, which may contribute to prove that there are infinite twin primes, has emerged last year. Yitang Zhang, of the University of New Hampshire in Durham, has published a proof in the Annals of Mathematics, showing that there is an infinite number of primes separated by no more than 70 million (which incidentally, in scientific notation, has the nicely symmetric form of 7 ∙ 107).

Whilst the study of prime numbers and their properties may look like mere recreational numerology, their role as building blocks of other integers turns them into a powerful practical tool for a procedure that most of us undergo on a daily basis. Which one? Juanjo Rué, from the Department of Mathematics and Computer Science of Berlin’s Freie Universität, will answer this and other questions during our Café Scientifique next week: “The building blocks of the numbers”. Come on by and listen to his talk!