Erdős Problem #10
Is there some k such that every large integer is the sum of a prime and at most k powers of 2?
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Open problems about additive bases
A set is an additive basis if every large integer is a sum of a bounded number of its elements. The questions ask how thin such a set can be and how the number of representations must behave (the Erdős–Turán conjecture is the best known).
Erdős Problem #11
Is every odd n > 1 the sum of a squarefree number and a power of 2?
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Erdős Problem #1145
Let A=1≤ a_1 < a_2 < ⋯ and B=1≤ b_1 < b_2 < ⋯ be sets of integers with a_n/b_n→ 1. If A+B contains all sufficiently large positive integers then is it true that limsup 1_Aast 1_B(n)=∞? A conjecture of Erdős and Sárközy.
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Erdős Problem #1192
Does there exist, for all r≥ 2, a basis A of order r (so that f_r(n)>0 for all large n) such that Σ_n≤ xf_r(n)^2 ≪ x for all x?
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Erdős Problem #28
If A ⊆ ℕ is such that A + A contains all but finitely many integers then limsup 1_A ∗ 1_A(n) = ∞.
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Erdős Problem #32
Does there exist a set A ⊆ ℕ such that |A ∩ 1, …, N| = o((log N)^2) and every sufficiently large integer can be written as p + a for some prime p and a ∈ A?
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Erdős Problem #326
Does there exist A = a_1 < a_2 < ⋯ ⊂ ℕ which is a minimal basis of order 2 (i.e. every large integer is the sum of 2 elements from A, and no proper subset of A has this property), such that lim_k→∞ a_k/k^2 = c for some c ≠ 0? Erdős and Graham conjectured a negative answer to this question [ErGr80].
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Erdős Problem #33
Let 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)?
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Erdős Problem #40
For what functions g(N) → ∞ is it true that lvert A∩ 1,…,Nrvert ≫ N^1/2/g(N) implies limsup 1_Aast 1_A(n)=∞?
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Erdős Problem #66
Is there and A ⊂ ℕ is such that lim_n→ ∞1_Aast 1_A(n)/log n exists and is ≠ 0?
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Erdős Problem #881
Let 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. Must there exist an infinite B ⊂ A such that A B is an additive basis of order k + 1?
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Erdős Problem #9
Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
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