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Mathematical Hypotheses

Mathematical hypotheses

continuum hypothesis | Riemann hypothesis

Statements

The continuum hypothesis states that there is no infinite set with a cardinal number between that of the "small" infinite set of integers and the "large" infinite set of real numbers.

The Riemann hypothesis posits that the nontrivial zeros of the Riemann zeta function ζ(s) all lie on the critical line Re(s) = 1/2.

Solutions

| solution
continuum hypothesis | undecidable

Formal statement

| formal statement
Riemann hypothesis | for all _(n, n element Z ∧ n!=0)Re(ρ_n) = 1/2

History

Associated equations

Re(ρ_n) = 1/2

Current evidence

Proved by Gödel and Cohen to be undecidable within Zermelo-Frankel set theory with or without the axiom of choice, but there is no consensus on whether this is a solution to the problem.

It has been verified that the first 1×10^13 nontrivial zeros of the zeta function lie on the critical line.
Conrey (1989) proved that at least 40% of the nontrivial zeros of the zeta function lie on the critical line.

Associated prizes

| prizes offered for solution | prizes awarded for solution
continuum hypothesis | | Paul Cohen received the Fields Medal in 1966 for showing that if set theory is consistent, then no contradiction would arise if the negation of the continuum hypothesis was added to set theory.
Riemann hypothesis | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. |

Common classes

mathematical hypotheses

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