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Erdős Problems

1,217 source-owned questions · 604 with a formal statement · searchable by statement, number, topic and source status.

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  • Open per source608
  • Resolved per source556
  • Other source status53
Exact formal statement available604 / 1,217
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Problems

27 Problems

SelectNumberQuestionOpen
#3If A⊂NA \subset \mathbb{N} has ∑n∈A1n=∞\sum_{n \in A}\frac 1 n = \infty, then must A contain arbitrarily long arithmetic progressions?Source says openFormalized
#139Erdős Problem 139: Let rk(N)r_k(N) be the size of the largest subset of 1,...,N{1,...,N} which does not contain a non-trivial kk-term arithmetic progression. Prove that rk(N)=o(N)r_k(N) = o(N).Source says provedFormalized
#140No statement retained — open to read what the source holdsSource says provedNo formal declaration
#141Let k≥3k≥3. Are there kk consecutive primes in arithmetic progression?Source says openFormalized
#142Prove an asymptotic formula for rk(N)r_k(N), the largest possible size of a subset of {1,…,N}\{1, \dots, N\} that does not contain any non-trivial kk-term arithmetic progression.Source says openFormalized
#160Estimate h(n)h(n) by finding a better lower bound.Source says openFormalized
#169No statement retained — open to read what the source holdsSource says openNo formal declaration
#176No statement retained — open to read what the source holdsSource says openNo formal declaration
#177No statement retained — open to read what the source holdsSource says openNo formal declaration
#179No statement retained — open to read what the source holdsSource says provedNo formal declaration
#190No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#192No statement retained — open to read what the source holdsSource says solved (Lean)No formal declaration
#194Let k≥3k\geq 3. Must any ordering of R\mathbb{R} contain a monotone kk-term arithmetic progression, that is, some x1<⋯<xkx_1 <\cdots < x_k which forms an increasing or decreasing kk-term arithmetic progression?Source says disproved (Lean)Formalized
#195What is the largest kk such that in any permutation of Z\mathbb{Z} there must exist a monotone kk-term arithmetic progression x1<⋯<xkx_1 < \cdots < x_k?Source says openFormalized
#196Must every permutation of N\mathbb{N}, contain a monotone 4-term arithmetic progression?Source says openFormalized
#197Can N\mathbb{N} be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?Source says openFormalized
#198The statement for which Baumgartner actually writes a proof.Source says disproved (Lean)Formalized
#199If A⊂RA\subset \mathbb{R} does not contain a 3-term arithmetic progression then must R\A\mathbb{R}\backslash A contain an infinite arithmetic progression?Source says disproved (Lean)Formalized
#200Does the longest arithmetic progression of primes in {1,…,N}\{1,\ldots,N\} have length o(log⁡N)o(\log N)?Source says openFormalized
#201No statement retained — open to read what the source holdsSource says openNo formal declaration
#219Are there arbitrarily long arithmetic progressions of primes? Solution: yes. Ref: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_Source says provedFormalized
#271No statement retained — open to read what the source holdsSource says openNo formal declaration
#272Let N≥1N\geq 1. What is the largest tt such that there are A1,…,At⊆{1,…,N}A_1,\ldots,A_t\subseteq \{1,\ldots,N\} with Ai∩AjA_i\cap A_j a non-empty arithmetic progression for all i≠ji\neq j?Source says openFormalized
#984No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1185No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1186No statement retained — open to read what the source holdsSource says openNo formal declaration
#1187No statement retained — open to read what the source holdsSource says solvedNo formal declaration