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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

31 Problems

SelectNumberQuestionOpen
#9Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?Source says openFormalized
#10Is there some kk such that every integer is the sum of a prime and at most kk powers of 22?Source says openFormalized
#11Is every odd n>1n > 1 the sum of a squarefree number and a power of 2?Source says openFormalized
#16Is the set of odd integers not of the form 2k+p2^k+p the union of an infinite arithmetic progression and a set of density 00?Source says disproved (Lean)Formalized
#28If A⊆NA ⊆ \mathbb{N} is such that A+AA + A contains all but finitely many integers then lim sup⁡1A∗1A(n)=∞\limsup 1_A ∗ 1_A(n) = \infty.Source says openFormalized
#29No statement retained — open to read what the source holdsSource says provedNo formal declaration
#31Given any infinite set A⊂NA\subset \mathbb{N} there is a set BB of density 00 such that A+BA+B contains all except finitely many integers.Source says proved (Lean)Formalized
#32Does there exist a set A⊆NA \subseteq \mathbb{N} such that ∣A∩{1,…,N}∣=o((log⁡N)2)|A \cap \{1, \ldots, N\}| = o((\log N)^2) and every sufficiently large integer can be written as p+ap + a for some prime pp and a∈Aa \in A?Source says openFormalized
#33Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. What is the smallest possible value of lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)?Source says openFormalized
#35No statement retained — open to read what the source holdsSource says provedNo formal declaration
#40For what functions g(N)→∞g(N) → \infty is it true that ∣A∩{1,…,N}∣≫N1/2g(N)\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)} implies lim sup⁡1A∗1A(n)=∞\limsup 1_A\ast 1_A(n)=\infty?Source says openFormalized
#66Is there and A⊂NA \subset \mathbb{N} is such that lim⁡n→∞1A∗1A(n)log⁡n\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n} exists and is ≠0\ne 0?Source says openFormalized
#221Is there a set A⊂NA\subset\mathbb{N} such that, for all large NN, ∣A∩{1,…,N}∣≪N/log⁡N\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N and such that every large integer can be written as 2k+a2^k+a for some k≥0k\geq 0 and a∈Aa\in A?Source says proved (Lean)Formalized
#326Let A⊂NA \subset \mathbb{N} be an additive basis of order 2.Source says openFormalized
#330Does there exist a minimal basis A⊂NA \subset \mathbb{N} with positive density such that, for any n∈An \in A, the (upper) density of integers which cannot be represented without using nn is positive?Source says proved (Lean)Formalized
#333Let A⊆NA\subseteq \mathbb{N} be a set of density zero. Does there exist a BB such that A⊆B+BA\subseteq B+B and ∣B∩{1,…,N}∣=o(N1/2)\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2}) for all large NN?Source says disproved (Lean)Formalized
#336No statement retained — open to read what the source holdsSource says openNo formal declaration
#337Let A⊆NA\subseteq \mathbb{N} be an additive basis (of any finite order) such that ∣A∩{1,…,N}∣=o(N)\lvert A\cap \{1,\ldots,N\}\rvert=o(N). Is it true that lim⁡N→∞∣(A+A)∩{1,…,N}∣∣A∩{1,…,N}∣=∞? \lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty? Source says disproved (Lean)Formalized
#338No statement retained — open to read what the source holdsSource says openNo formal declaration
#339No statement retained — open to read what the source holdsSource says provedNo formal declaration
#358When An=nA_n = n, the function ff defined above counts the number of odd divisors of nn.Source says provedFormalized
#868Let AA be an additive basis of order 22, let f(n)f(n) denote the number of ways in which nn can be written as the sum of two elements from AA. If f(n)→∞f(n) \to \infty as n→∞n \to \infty, then must AA contain a minimal additive basis of order 22?Source says solvedFormalized
#869No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#870No statement retained — open to read what the source holdsSource says openNo formal declaration
#871Let AA be an additive basis of order 22, and suppose 1A∗1A(n)→∞1_A\ast 1_A(n)\to \infty as n→∞n\to \infty. Can AA be partitioned into two disjoint additive bases of order 22?Source says disproved (Lean)Formalized
#880No statement retained — open to read what the source holdsSource says provedNo formal declaration
#881Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A \ B is not a basis of order k.Source says openFormalized
#1145Let A={1≤a1<a2<⋯ }A=\{1\leq a_1 < a_2 < \cdots\} and B={1≤b1<b2<⋯ }B=\{1\leq b_1 < b_2 < \cdots\} be sets of integers with an/bn→1a_n/b_n\to 1.Source says openFormalized
#1147No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#1192Does there exist, for all r≥2r\geq 2, a basis AA of order rr (so that fr(n)>0f_r(n)>0 for all large nn) such that ∑n≤xfr(n)2≪x\sum_{n\leq x}f_r(n)^2 \ll x for all xx?Source says openFormalized
#1194No statement retained — open to read what the source holdsSource says openNo formal declaration