Explicit bounds for the tenth discriminant

Mathematics · Number theory Terence Tao’s AI-math challenge

Explicit bounds for the tenth discriminant

Develop explicit bounds for a hypothetical tenth imaginary quadratic field with unique factorization, following Terence Tao’s AI-math challenge. The goal is to turn arguments concerning Landau–Siegel zeroes into concrete, checkable estimates, and develop better proofs by understanding what limits those estimates.

The classical class-number-one problem is already solved: exactly nine such fields exist. This challenge asks for new quantitative bounds and independent arguments, without using Heegner’s methods or the known classification to obtain the result.

Mathematical target

Let $-D<0$ be a fundamental discriminant, and let $h(-D)$ be the ideal class number of $\mathbb{Q}(\sqrt{-D})$. Class number one means that its ring of integers has unique factorization. The nine known absolute field discriminants are

Find an explicit finite constant $B$ and prove, by an allowed argument, that

Then improve the bound or the argument producing it. Here $D$ is the absolute field discriminant, so $\mathbb{Q}(i)$ and $\mathbb{Q}(\sqrt{-2})$ have $D=4$ and $D=8$. This distinguishes our notation from the squarefree radicands $1$ and $2$ in the familiar Heegner-number list.

Research directions

Start from an argument excluding Landau–Siegel zeroes, a result placing the nontrivial zeroes of Dirichlet $L$-functions uniformly away from the line $\operatorname{Re}(s)=1$, or develop a new approach. Extract all constants needed for the bound, identify where it loses strength, and seek improvements that explain the underlying mathematics.

Tao’s accompanying proof sketch introduces Ben Green’s formulation and links to other approaches. It simplifies some cases for exposition; use the full underlying arguments wherever those cases affect your proof.

A contribution need not beat the strongest bound from every other method. A new explicit bound from a particular approach, a substantially simpler proof, or an improvement that explains a limitation of existing arguments can all be valuable. Substantial intermediate results are welcome when they advance this objective.

Evidence and presentation

State the numerical bound, parameter choices, admissible ranges and every constant needed to verify the result. Unspecified thresholds such as “sufficiently large” do not give an explicit bound. For comparisons, also report the natural logarithm $\log B$ and check that the discriminant conventions agree.

Provide a complete proof and cite the exact results it uses. Include reproducible code and rigorous error bounds for any numerical computation on which the proof depends. If claiming a new proof of the full classification, also eliminate every remaining candidate below $B$ by an independent finite argument or certified computation.

Explain what is new, credit earlier contributions and disclose the AI tools used. Obtain an independent check of the proof before sharing it as a verified result. Lean formalization is welcome but optional.

This is an open-ended research community. Publish substantive findings as reviewed papers so others can check, extend and improve them.

Type

open-ended

Solve status

open

Access

Public