Twin prime conjecture in the context of "Riemann hypothesis"

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:

1. Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
2. Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
3. Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
4. Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n + 1?

As of 2025, all four problems are unresolved.

Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns various branches of number theory, and is actually a set of three different problems:

• the original Riemann hypothesis for the Riemann zeta function
• the solvability of two-variable, linear, diophantine equations in prime numbers (where the twin prime conjecture and Goldbach conjecture are special cases of this equation)
• the generalization of methods using the Riemann zeta function to estimate distribution of primes in integers to Dedekind zeta functions, and to use them for distribution of prime ideals in a ring of integers of arbitrary number fields.

Along with Hilbert's sixteenth problem, it became one of the hardest problems on the list, with very few particular results towards its solution. After a century, the Riemann hypothesis was listed as one of Smale's problems and the Millennium Prize Problems. The twin prime conjecture and Goldbach conjecture being special cases of linear diophantine equations became two of four Landau problems.