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Divisibility and PrimesThe Riemann Hypothesis

However, as you can see above, there is still a significant error between the actual number of primes, and Gauss’s approximation. In 1859, the mathematician Bernhard Riemann

Bernhard Riemann (1826 – 1866) was a German mathematician working in the fields of analysis and number theory. He came up with the first rigorous definition of integration, studied differential geometry which laid the foundation for general relativity, and made groundbreaking discoveries regarding the distribution of prime numbers.

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discovered an approximation that looked much better, but he wasn’t able to prove that it would always work. His idea became known as the Riemann Hypothesis.

Hundreds of mathematicians have tried to prove Riemann’s hypothesis, but all without success. It is often considered one of the most difficult and most important unsolved problems in mathematics. In 2000, the Clay Mathematics Institute named it one of seven Millennium Prize Problems

The seven Millennium prize problems are some of the most difficult open problems in Mathematics. They were listed in 2000 by the Clay Mathematics Institute, and each hold a $1m reward:

  • P vs NP
  • The Hodge conjecture
  • The Poincaré conjecture
  • The Riemann hypothesis
  • Yang-Mills existence and mass gap
  • Navier-Stokes existence and smoothness
  • The Birch and Swinnerton-Dyer conjecture

So far, just one of the problems, the Poincaré conjecture, has been solved. However the mathematician who solved it, Grigori Perelman, declined the award.

and promised $1,000,000 to any mathematician who solves it.

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