In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the nontrivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics.
What would be the philosophical implications of a proof of the Riemann hypothesis?
Over two thousand years ago the Greeks proved that every number can be written as a product of prime numbers. A fast and efficient way to find which prime numbers have been used to build up other numbers has eluded mathematicians ever since. What we are missing is a mathematical counterpart of chemical spectroscopy, which tells chemists which elements of the Periodic Table make up a chemical compound. A discovery of a mathematical analogue that would crack numbers into their constituent primes would earn its creator more than just academic acclaim.
In 1903, Cole's calculation was regarded as an interesting mathematical curiosity - the standing ovation he received was in recognition of his extraordinary hard labour rather than any intrinsic importance the problem had. Such number-cracking is no longer a Sunday afternoon pastime but lies at the heart of modern code-breaking. Mathematicians have devised a way to wire this difficult problem of cracking numbers into the codes that protect the world's finances on the Internet. This innocent sounding task is sufficiently tough for numbers with 100 digits that banks and e-commerce are prepared to stake the security of their financial transactions on the impossible long time it takes - at present - to find the prime factors. At the same time, these new mathematical codes have been used to solve a problem that dogged the world of cryptography.
Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example 512 bit primes are frequently used for RSA and 1024 bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.
Wouldn't the ability to crack the codes that protect the information of Governments, Banks and Corporations, and every individuals personal information necessarily have Ethical implications as a possible weapon? (theoretically a person could drastically alter the course of the world using a smartphone on their coffee break) And also wouldn't the ability to predict the distribution of the primes necessarily have philosophical implications regarding pure mathematics and philosophy of mathematics?