Category:Hilbert 23

Hilbert 23

The Hilbert 23 is a list of $23$ at-the-time unsolved problems in mathematics published by David Hilbert during $1900$.

They are as follows:

1: The Continuum Hypothesis

There is no set whose cardinality is strictly between that of the integers and the real numbers.

2: Consistency of Axioms of Mathematics

Proof that the axioms of mathematics are consistent.

Consistency of Axioms of Mathematics

3: Finite Dissection of Polyhedra

Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

Finite Dissection of Polyhedra

4: Construction of all Metrics where Lines are Geodesics

Construct all metrics where lines are geodesics.

Construction of all Metrics where Lines are Geodesics

5: Whether Continuous Groups are Differential Groups

Are continuous groups automatically differential groups?

Whether Continuous Groups are Differential Groups

6: Axiomatize all of Physics

Axiomatize all of physics.

Mathematical treatment of the axioms of physics:

$(1): \quad$ An axiomatic treatment of probability with limit theorems for foundation of statistical physics

$(2): \quad$ The rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua".

7: The Gelfond-Schneider Theorem

Let $\alpha$ and $\beta$ be algebraic numbers (possibly complex) such that $\alpha \notin \set {0, 1}$.

Let $\beta$ be irrational.

Then any value of $\alpha^\beta$ is transcendental.

8a: The Riemann Hypothesis

All the nontrivial zeroes of the analytic continuation of the Riemann zeta function $\zeta$ have a real part equal to $\dfrac 1 2$.

8b: The Goldbach Conjecture

Every even integer greater than $2$ is the sum of two primes.

8c: The Twin Prime Conjecture

It is conjectured that there exist infinitely many pairs of twin primes: that is, primes which differ by $2$.

9: General Reciprocity Theorem in Algebraic Number Field

Find the most general law of the Reciprocity Theorem in any algebraic number field.

General Reciprocity Theorem in Algebraic Number Field

10: Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution

There is no algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

11: Quadratic Forms with Algebraic Numerical Coefficients

Solving quadratic forms with algebraic numerical coefficients.

Quadratic Forms with Algebraic Numerical Coefficients

12: Extension of Kronecker-Weber Theorem to any base Number Field

Extend the Kronecker-Weber Theorem on abelian extensions of the rational numbers to any base number field.

Extension of Kronecker-Weber Theorem to any base Number Field

13: Solution of 7th Degree Equations using Two Parameter Functions

Solve all $7$th degree equations using functions of two parameters.

Solution of 7th Degree Equations using Two Parameter Functions

18b: Kepler's Conjecture (Densest Sphere Packing)

The densest packing of identical spheres in space is obtained when the spheres are arranged with their centers at the points of a face-centered cubic lattice.

This obtains a density of $\dfrac \pi {3 \sqrt 2} = \dfrac \pi {\sqrt {18} }$:

$\dfrac \pi {\sqrt {18} } = 0 \cdotp 74048 \ldots$