Problem Catalog

19 problems across 5 fields of mathematics

Number Theory#728
AISOLVED

Erdos Problem #728

How large can the gap k = a + b - n be, given that a!b! divides n!k!?

a! b!∣n! (a+b−n)!withC1log⁡n<a+b−n<C2log⁡na!\, b! \mid n!\, (a+b-n)! \quad\text{with}\quad C_1 \log n < a+b-n < C_2 \log n
*****
1975
GPT-5.2 Pro, Aristotle (Harmonic) — Jan 6, 2026
Combinatorics#125
AISOLVED

Erdos Problem #125

A combinatorial problem on set systems proposed by Erdos.

DeepMind prover: full solution in Lean\text{DeepMind prover: full solution in Lean}
*****
1936
DeepMind Prover Agent — Mar 30, 2026
Number Theory#205
AISOLVED

Erdos Problem #205

A number-theoretic problem resolved with a formal Lean proof by AI.

Full AI solution (Lean), Jan 2026\text{Full AI solution (Lean), Jan 2026}
*****
1950
Aristotle, GPT-5.2 Thinking — Jan 10, 2026
Number Theory
OPEN

Erdos-Straus Conjecture

For every integer n >= 2, the fraction 4/n can be expressed as a sum of three unit fractions.

4n=1x+1y+1z\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}
*****
1948
Number Theory
OPEN

Erdos Conjecture on Arithmetic Progressions

If A is a set of positive integers whose reciprocals diverge, then A contains arithmetic progressions of every finite length.

∑n∈A1n=∞  ⟹  A⊇AP of length k,  ∀k\sum_{n \in A} \frac{1}{n} = \infty \;\Longrightarrow\; A \supseteq \text{AP of length } k,\; \forall k
*****
1973
Number Theory
SOLVED

Erdos-Kac Theorem

The number of distinct prime factors of n is approximately normally distributed.

ω(n)−ln⁡ln⁡nln⁡ln⁡n→dN(0,1)\frac{\omega(n) - \ln\ln n}{\sqrt{\ln\ln n}} \xrightarrow{d} \mathcal{N}(0,1)
*****
1940
Combinatorics
SOLVED

Erdos-Ko-Rado Theorem

Maximum size of an intersecting family of r-element subsets of an n-element set.

∣F∣≤(n−1r−1)for n≥2r|\mathcal{F}| \leq \binom{n-1}{r-1} \quad \text{for } n \geq 2r
*****
1961
Combinatorics
SOLVED

Erdos-Szekeres Theorem

Any sequence of more than (r-1)(s-1) distinct reals contains a monotone subsequence of length r or s.

n>(r−1)(s−1)  ⟹  ∃ monotone subsequence of length r or sn > (r-1)(s-1) \implies \exists\, \text{monotone subsequence of length } r \text{ or } s
*****
1935
Combinatorics
SOLVED

Erdos Discrepancy Problem

For any infinite +/-1 sequence, partial sums along arithmetic progressions are unbounded.

sup⁡N,d≥1∣∑k=1Nxkd∣=∞\sup_{N,d \geq 1} \left|\sum_{k=1}^{N} x_{kd}\right| = \infty
*****
1932
Graph Theory
SOLVED

Erdos-Gallai Theorem

Characterizes which integer sequences are degree sequences of simple graphs.

∑i=1kdi≤k(k−1)+∑i=k+1nmin⁡(di,k)\sum_{i=1}^{k} d_i \leq k(k-1) + \sum_{i=k+1}^{n} \min(d_i, k)
*****
1960
Probability
SOLVED

Erdos-Renyi Random Graph Model

In G(n,p), a sharp phase transition occurs at p = 1/n: a giant component emerges.

G(n,p):  p=cn⇒{small componentsc<1giant componentc>1G(n,p):\; p=\frac{c}{n} \Rightarrow \begin{cases} \text{small components} & c<1 \\ \text{giant component} & c>1 \end{cases}
*****
1959
Graph Theory
SOLVED

Erdos-Faber-Lovasz Conjecture

The union of n complete graphs sharing at most one vertex each is n-colorable.

χ ⁣(⋃i=1nKn(i))≤n\chi\!\left(\bigcup_{i=1}^{n} K_n^{(i)}\right) \leq n
*****
1972
Number Theory
SOLVED

Erdos-Ginzburg-Ziv Theorem

Among any 2n-1 integers, there exist n whose sum is divisible by n.

∀ a1,…,a2n−1∈Z,  ∃ S⊂[2n−1],  ∣S∣=n:∑i∈Sai≡0(modn)\forall\, a_1,\ldots,a_{2n-1} \in \mathbb{Z},\; \exists\, S \subset [2n-1],\; |S|=n : \sum_{i\in S} a_i \equiv 0 \pmod{n}
*****
1961
Number Theory
OPEN

Erdos-Turan Conjecture on Additive Bases

If B is an additive basis of order 2, the representation function is unbounded.

lim sup⁡n→∞rB,2(n)=∞\limsup_{n\to\infty} r_{B,2}(n) = \infty
*****
1941
Geometry
SOLVED

Erdos Distinct Distances Problem

n points in the plane determine at least Omega(n/log n) distinct distances.

g(n)=Ω ⁣(nlog⁡n)g(n) = \Omega\!\left(\frac{n}{\log n}\right)
*****
1946
Number Theory#457
AISOLVED

Erdos Problem #457

A number-theoretic problem resolved autonomously by AI with formal verification.

Full AI solution (Lean), Mar 2026\text{Full AI solution (Lean), Mar 2026}
*****
1980
Aristotle, GPT-5.2 Pro — Mar 2, 2026
Combinatorics#846
AISOLVED

Erdos Problem #846

A combinatorial problem independently solved by two AI labs.

Full solution (Lean), Feb 2026\text{Full solution (Lean), Feb 2026}
*****
1980
DeepMind Prover Agent — Feb 21, 2026
Combinatorics#347
AISOLVED

Erdos Problem #347

A combinatorial problem solved through extensive human-AI collaboration.

Full solution (Lean), Feb 2026\text{Full solution (Lean), Feb 2026}
*****
1970
Aristotle, Claude Opus, Codex, GPT — Oct 2025 - Feb 2026
Combinatorics#1051
AISOLVED

Erdos Problem #1051

A combinatorial problem solved by the Aletheia AI system.

Full solution (Lean), Jan 2026\text{Full solution (Lean), Jan 2026}
*****
2000
Aletheia — Jan 29, 2026