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from every problem. erdosproblems.com no longer shows statuses (Bloom's reasons); this is the database's record, not our verdict.
908 of 1221 statements are formalized in Lean (Formal Conjectures) · 107 carry an Erdős prize · 3 are independent of ZFC and 9 more can't be proved, or can't be disproved, from ZFC alone · “open” counts the computable and Lean-checked ones too.
Let \((S_n)_{n \ge 1}\) be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function \(f(z)\) such that, for all \(n \ge 1\), there exists some \(k_n \ge 0\) such…
Let \(\phi(n)\) be the Euler totient function and \(\phi_k(n)\) be the iterated \(\phi\) function, so that \(\phi_1(n)=\phi(n)\) and \(\phi_k(n)=\phi(\phi_{k-1}(n))\). Let \[f(n) = \min\{k : \phi_k(n)=1\}.\] Does \(f(n)/\log n\) have a…
Let \(σ_1(n) = σ(n)\), the sum of divisors function, and \(σ_k(n) = σ(σ_{k-1}(n))\). Is it true that \(\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞\)? This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the…
Let \(σ_1(n)=σ(n)\), the sum of divisors function, and \(σ_k(n) = σ(σ_{k-1}(n))\). Is it true that, for every \(m, n ≥ 2\), there exist some \(i, j\) such that \(σ_i(m) = σ_j(n)\)?
Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., { z | ∃ k, iteratedDeriv (n k) f z = 0 } is dense.
OPENPRIZE $500SKIPPEDnumber theory · iterated functions
Collatz conjecture
The Collatz conjecture states that for any positive integer \(n\), there exists a natural number \(m\) such that the \(m\)-th term of the sequence is 1.
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