Erdős problems
Paul Erdős posed more than 1,200 problems across combinatorics, number theory and graph theory. The Qvanta® Erdős Problems Research Program attacks selected ones with AI-assisted formal proof.
Qvanta® active formal targets
An open Erdős conjecture on the existence of distinct integer configurations for every n > 2, with equivalent formulations in terms of prime congruences and unit-fraction representations (see Bloom–Elsholtz and the catalog for the full statement).
Qvanta® holds partial, machine-checkable progress on sub-lemmas and certified special cases toward this problem. Full formalization and bound verification remain in active development.
Number Theory · T. F. Bloom, Erdős Problem #242, erdosproblems.com/242Let X be a finite set of size n and H(n) be minimal such that some f mapping subsets of X into X satisfies { f(A) : A ⊆ Y } = X for every Y ⊆ X with |Y| ≥ H(n). Prove that H(n) − log₂ n → ∞.
Qvanta® has partial proofs and formally verified intermediate bounds building on the Erdős–Hajnal framework and subsequent refinements. Work continues toward a complete machine-checked argument.
Combinatorics · T. F. Bloom, Erdős Problem #624, erdosproblems.com/624Attacking open problems at the boundary of mathematical knowledge
Qvanta® selects targeted problems from the Erdős Problems catalog and attacks them with AI-assisted formal proof infrastructure. The output is verified proof: every claim compiled and machine-checked where complete, and certified partial progress where the full statement remains open.
Formal proof. Agentic search. No hallucinations.
Language models propose proof paths, tactics and candidate lemmas; a formal proof assistant compiles or rejects every step. No proof is accepted that the verifier does not compile.
Foundational mathematics as a security primitive
The combinatorial bounds and number-theoretic identities in Erdős problems are the same structures that govern cryptographic hardness and post-quantum security arguments.
All 1221 Erdős problems
Each tile links to the problem on erdosproblems.com. Hover or long-press a tile for its exact status and topics.
Proved, disproved, otherwise solved, or shown independent of ZFC. “· Lean” means the solution is formalized in Lean.
Reduced to a finite computation, shown unprovable or undisprovable in ZFC, or with a Lean proof not yet reviewed by a human.
Appears completely open.