Erdős Problem #148
Let F(k) be the number of solutions to 1= 1/n_1+⋯+1/n_k, where 1≤ n_1<⋯<n_k are distinct integers. Find good estimates for F(k).
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Open problems about unit fractions (Egyptian fractions)
Writing numbers as sums of distinct fractions 1/n: which denominators suffice, how many terms are needed, and what happens when the denominators are restricted. Includes problems around the Erdős–Straus conjecture.
Erdős Problem #282
Let A⊆ ℕ be an infinite set and consider the following greedy algorithm for a rational x∈ (0,1): choose the minimal n∈ A not used so far such that n≥ 1/x and repeat with x replaced by x-1/n.
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Erdős Problem #287
Let k≥2. Is it true that, for any distinct integers 1 < n_1 < ⋯ < n_k such that Σ_i=1^k 1/n_i = 1, we must have max(n_i+1 - n_i) ≥ 3?
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Erdős Problem #288
Is it true that there are only finitely many pairs of intervals I_1, I_2 such that Σ_n_1 ∈ I_1 1/n_1 + Σ_n_2 ∈ I_2 1/n_2 ∈ ℕ?
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Erdős Problem #289
Is it true that, for all sufficiently large k, there exist finite intervals I_1, dotsc, I_k ⊂ ℕ, distinct, not overlapping or adjacent, with |I_i| ≥ 2 for 1 ≤ i ≤ k such that 1 = Σ_i=1^k Σ_n ∈ I_i 1/n?
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Erdős Problem #291
Let n≥ 1 and define L_n to be the least common multiple of 1,…,n and a_n by Σ_1≤ k≤ n1/k=a_n/L_n. Is it true that (a_n,L_n)=1 occurs for infinitely many n?
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Erdős Problem #295
Let k(N) denote the smallest k such that there exists N ≤ n_1 < ⋯ < n_k with frac 1 n_1 + ... + frac 1 n_k = 1 Is it true that lim_N → ∞ k(N) - (e - 1)N = ∞?
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Erdős Problem #302
Let f(N) be the size of the largest A⊆ 1,…,N such that there are no solutions to 1/a= 1/b+1/c with distinct a,b,c∈ A? Estimate f(N). The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].
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Erdős Problem #304
Is it true that N(b) ≪ log log b?
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Erdős Problem #306
Let frac a b∈ ℚ_>0 with b squarefree. Are there integers 1 < n_1 < … < n_k, each the product of two distinct primes, such that a/b=1/n_1+⋯+1/n_k?
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Erdős Problem #307
Are there two finite set of primes P and Q such that 1 = ( Σ_p ∈ P 1/p ) ( Σ_q ∈ Q 1/q ) ? Asked by Barbeau [Ba76]. [Ba76] Barbeau, E. J., _Computer challenge corner: Problem 477: A brute force program._
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Erdős Problem #312
Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with Σ_n ∈ A 1/n > K there exists some S ⊆ A such that 1 - exp(-(c*K)) < Σ_n ∈ S 1/n ≤ 1?
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Erdős Problem #313
Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: Σ_p ∈ P 1/p = 1 - 1/m?
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Erdős Problem #317
Is there some constant c>0 such that for every n≥ 1 there exists some δ_k∈ -1,0,1 for 1≤ k≤ n with 0< lvert Σ_1≤ k≤ nδ_k/krvert < c/2^n?
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Erdős Problem #319
What is the size of the largest A⊆1, …, N such that there is a function δ : A → -1, 1 such that Σ_n∈ A δ n/n = 0 and Σ_n∈ A'δ n/n ≠ 0 for all non-empty A'subsetneq A.
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