# Erdős problems > 13 AI-made results on Erdős problems, each explained in plain English. Questions first asked or popularised by Paul Erdős, the prolific 20th-century… Canonical: https://closedform.nishachay.tech/topic/erdos-problems/ Last updated: 2026-10-14 Questions first asked or popularised by Paul Erdős, the prolific 20th-century mathematician who posed hundreds of problems, often with cash prizes. 13 AI-made results on Erdős problems, from 16 papers in the OpenAI math collection, each explained in plain English. 11 have a formal Lean proof listed; 2 are claimed in a paper and not yet machine-checked. They span 3 fields, led by combinatorics (8), number theory (4) and real and complex analysis (1). The oldest question here dates back to 1936: [Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds](https://closedform.nishachay.tech/e/openai-math-2026-159/). Rated Landmark: [Planar distinct distances and unit-distance bounds](https://closedform.nishachay.tech/e/openai-math-2026-167/). ## Results (13) - [Independent largest prime factors of consecutive integers](https://closedform.nishachay.tech/e/openai-math-2026-012/) - Entry 012 · The largest prime factors of n and n + 1 act like independent dice, so each wins half the time. (Formal proof listed) - [An asymptotic formula for the number of totients](https://closedform.nishachay.tech/e/openai-math-2026-024/) - Entry 024 · Counting totient values up to x gets an exact leading formula, and doubling x doubles the count in the limit. (Formal proof listed) - [Short Egyptian fractions](https://closedform.nishachay.tech/e/openai-math-2026-025/) - Entry 025 · Every fraction below 1 is a sum of about log log b distinct unit fractions, as Erdős conjectured. (Formal proof listed) - [Positive lower density of large prime gaps](https://closedform.nishachay.tech/e/openai-math-2026-026/) - Entry 026 · For any size bar, a fixed share of prime gaps clears it, all the way out. (Formal proof listed) - [The geometric case of the Erdős similarity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-084/) - Entry 084 · A set covering almost all of [0, 1] can dodge every copy of a geometric list like 1/2, 1/4, 1/8, … (Formal proof listed) - [Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds](https://closedform.nishachay.tech/e/openai-math-2026-159/) - Entry 159 · Any set of whole numbers whose reciprocals add to infinity holds evenly spaced runs of every length. (Formal proof listed) - [Superexponential van der Waerden numbers](https://closedform.nishachay.tech/e/openai-math-2026-160/) - Entry 160 · Proves van der Waerden numbers grow faster than any exponential, answering a question of Erdős for two colors. (Formal proof listed) - [The higher-dimensional Erdős distinct-distances conjecture](https://closedform.nishachay.tech/e/openai-math-2026-166/) - Entry 166 · Any n points in d ≥ 3 dimensions span at least a constant times n^(2/d) distinct distances, the paper claims. (Claimed, not checked) - [Planar distinct distances and unit-distance bounds](https://closedform.nishachay.tech/e/openai-math-2026-167/) - Entry 167 · Proves almost every point in a large planar set sees nearly n distances, and lowers the unit-distance exponent 4/3. (Formal proof listed) - [The hypercube Ramsey conjecture](https://closedform.nishachay.tech/e/openai-math-2026-171/) - Entry 171 · Two-coloring the links among a constant times 2ⁿ points always gives a one-color n-dimensional cube, the paper claims. (Claimed, not checked) - [The Erdős–Gallai cycle-decomposition conjecture](https://closedform.nishachay.tech/e/openai-math-2026-181/) - Entry 181 · Every network on n points splits into at most a fixed multiple of n loops and single links. (Formal proof listed) - [Correspondence coloring with a fixed forbidden subgraph](https://closedform.nishachay.tech/e/openai-math-2026-184/) - Entry 184 · Networks with no fixed-size clique have larger independent sets and need fewer colors, by a log factor. (Formal proof listed) - [Cycle–clique Ramsey numbers](https://closedform.nishachay.tech/e/openai-math-2026-189/) - Entry 189 · Exact Ramsey numbers for a cycle versus a clique: R(C_m, K_n) = (m − 1)(n − 1) + 1 when m ≥ n ≥ 3. 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