number-theory Jan 29, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #1051.” The registry entry and named primary source contain the available statement and scope.
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number-theory Jun 22, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #865.” The registry entry and named primary source contain the available statement and scope.
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geometry-topology Feb 25, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
VibeMathed records this result as “Erdős Problem #846.” The registry entry and named primary source contain the available statement and scope.
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quantum-information-computing Aug 27, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → disproved
For two qubits, the paper considers
ρ = 1 2 ∣ Φ + ⟩ ⟨ Φ + ∣ + 1 2 ∣ 01 ⟩ ⟨ 01 ∣ \rho=\frac12|\Phi^+\rangle\langle\Phi^+|+\frac12|01\rangle\langle01| ρ = 2 1 ∣ Φ + ⟩ ⟨ Φ + ∣ + 2 1 ∣01 ⟩ ⟨ 01∣ ,
where ∣ Φ + ⟩ = ( ∣ 00 ⟩ + ∣ 11 ⟩ ) / 2 |\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2 ∣ Φ + ⟩ = ( ∣00 ⟩ + ∣11 ⟩) / 2 . This rank-2 state has no global supporting affine functional for Entanglement of Formation.
Setting ρ t = ( 1 − t ) ρ + t ∣ 10 ⟩ ⟨ 10 ∣ \rho_t=(1-t)\rho+t|10\rangle\langle10| ρ t = ( 1 − t ) ρ + t ∣10 ⟩ ⟨ 10∣ , Wootters’ formula gives
C ( ρ t ) = 1 2 − 2 t + O ( t ) C(\rho_t)=\frac12-\sqrt{2t}+O(t) C ( ρ t ) = 2 1 − 2 t + O ( t ) .
Consequently,…
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analysis Jul 1, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Among all nonconstant monic polynomials f f f whose roots lie in [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] , determine inf f ∣ { x ∈ R : ∣ f ( x ) ∣ < 1 } ∣ \inf_f |\{x \in \mathbb{R} : |f(x)| < 1\}| inf f ∣ { x ∈ R : ∣ f ( x ) ∣ < 1 } ∣ .
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probability-statistics Jul 27, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → disproved
Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a finite-energy Wigner counterexample and a one-parameter family of Gaussian-regularised Bernoulli entry laws answer both questions negatively.
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algebra Jul 22, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Dogon, Levit and Vigdorovich asked for an explicit upper bound on the stability radius of an infinitely presented group. The lamplighter group provides the first: explicit polynomial bounds on both its Hilbert-Schmidt stability rate and its stability radius, obtained through approximately invariant measures and an effective marker construction.
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number-theory Mar 30, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
VibeMathed records this result as “Erdős Problem #125.” The registry entry and named primary source contain the available statement and scope.
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number-theory Apr 14, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #258.” The registry entry and named primary source contain the available statement and scope.
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probability-statistics Jul 3, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
uniqueness proved for stable Harrison-Reiman systems with a nonsingular M-matrix reflection; an infinite-dimensional obstruction is shown in the larger completely-S class
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number-theory Apr 25, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #38.” The registry entry and named primary source contain the available statement and scope.
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combinatorics Jun 26, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
A subset A A A of the pointwise-ordered cube [ 0 , 1 ] n [0,1]^n [ 0 , 1 ] n is a k k k -antichain when it meets every chain in at most k k k points. The conjecture concerns the largest possible ( n − 1 ) (n-1) ( n − 1 ) -dimensional Hausdorff measure of such a set; it is settled here, following work of Janzer.
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number-theory Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
If A ⊆ N A \subseteq \mathbb{N} A ⊆ N has unbounded dyadic-shell counts and ∑ n ∈ A ∥ θ n ∥ = ∞ \sum_{n \in A} \|\theta n\| = \infty ∑ n ∈ A ∥ θ n ∥ = ∞ for every 0 < θ < 1 0 < \theta < 1 0 < θ < 1 , must A A A be complete - is every sufficiently large integer a sum of distinct elements of A A A ?
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combinatorics May 7, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
a new density-degree inequality gives δ(G) ≤ (3/10 + o(1))|V(G)|, improving 0.328; existence of a linear construction remains open
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number-theory Jan 5, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
VibeMathed records this result as “Erdős Problem #871.” The registry entry and named primary source contain the available statement and scope.
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number-theory Apr 24, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #330.” The registry entry and named primary source contain the available statement and scope.
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probability-statistics May 15, 2026 Significance 10/100 Registry: expert verified
Prior state unknown → proved
For the switch-walk-switch lamplighter walk on Z 2 ≀ T d \mathbb{Z}_2 \wr T_d Z 2 ≀ T d , prove the sharp asymptotic p 2 n ( e , e ) = ρ d 2 n exp [ − ( π 2 ( log ( d − 1 ) ) 2 + o ( 1 ) ) n log 2 n ] p_{2n}(e,e) = \rho_d^{2n} \exp[-(\pi^2 (\log(d-1))^2 + o(1)) \frac{n}{\log^2 n}] p 2 n ( e , e ) = ρ d 2 n exp [ − ( π 2 ( log ( d − 1 ) ) 2 + o ( 1 )) l o g 2 n n ] with ρ d = 2 d − 1 d \rho_d = \frac{2\sqrt{d-1}}{d} ρ d = d 2 d − 1 .
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number-theory Jun 27, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
resolved under an explicit formalization of 'reasonable', not in full generality
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combinatorics Jul 23, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
corank 3; the general conjecture remains open, corank 2 being McConville's case
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number-theory May 2, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → disproved
A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open
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number-theory Jun 19, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
If a / b ∈ Q > 0 a/b \in \mathbb{Q}_{>0} a / b ∈ Q > 0 and b b b is squarefree, can a / b a/b a / b always be written as a finite sum of reciprocals of distinct products of two distinct primes?
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algorithms-optimization Jul 23, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → disproved
restricted planar four-terminal case
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analysis Jun 21, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
VibeMathed records this result as “Erdős Problem #1197.” The registry entry and named primary source contain the available statement and scope.
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algorithms-optimization Apr 4, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
For smooth convex-concave min-max problems, can anchored gradient descent-ascent be scheduled so that its exact last-iterate squared-gradient residual is O ( 1 / t ) O(1/t) O ( 1/ t ) , closing the gap left by the 2019 analysis?
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number-theory Feb 5, 2026 Significance 10/100 Registry: site confirmed
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #851.” The registry entry and named primary source contain the available statement and scope.
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combinatorics Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
the literal wording is refuted; the intended variant remains open
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geometry-topology Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → disproved
natural readings of the ambiguous historical statement are disproved
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algebra Aug 19, 2026 Significance 10/100 Registry: lean checked
Prior state unknown → proved
For the Kasami APN function F ( x ) = x 4 k − 2 k + 1 F(x) = x^{4^k - 2^k + 1} F ( x ) = x 4 k − 2 k + 1 on G F ( 2 n ) \mathrm{GF}(2^n) GF ( 2 n ) with gcd ( k , n ) = 1 \gcd(k, n) = 1 g cd( k , n ) = 1 , the conjecture asserts that for Δ = { F ( b ) + F ( b + 1 ) + 1 } \Delta = \{F(b) + F(b+1) + 1\} Δ = { F ( b ) + F ( b + 1 ) + 1 } and all distinct nonzero v 1 , v 2 v_1, v_2 v 1 , v 2 , the number of triples in Δ 3 \Delta^3 Δ 3 with v 1 x + v 2 y + ( v 1 + v 2 ) z = 0 v_1 x + v_2 y + (v_1 + v_2) z = 0 v 1 x + v 2 y + ( v 1 + v 2 ) z = 0 is exactly 2 2 n − 3 2^{2n-3} 2 2 n − 3 . Proved for k m o d n ∈ { 1 , 2 , n − 2 , n − 1 } k \bmod n \in \{1, 2, n-2, n-1\} k mod n ∈ { 1 , 2 , n − 2 , n − 1 } and verified exhaustively for n ≤ 13 n \le 13 n ≤ 13 ; the general case remains open.
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number-theory Mar 7, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
VibeMathed records this result as “Erdős Problem #650.” The registry entry and named primary source contain the available statement and scope.
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number-theory Jan 6, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
Whether there are infinitely many integers a , b , n a, b, n a , b , n with a , b ≥ ε n a, b \ge \varepsilon n a , b ≥ ε n such that a ! ⋅ b ! a!\cdot b! a ! ⋅ b ! divides n ! ⋅ ( a + b − n ) ! n!\cdot(a+b-n)! n ! ⋅ ( a + b − n )! while a + b a+b a + b exceeds n n n by more than C ⋅ log n C\cdot\log n C ⋅ log n .
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differential-equations Jul 27, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → disproved
the named open question is answered negatively; the paper's positive theory goes further
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combinatorics Jul 28, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
girth >= 5 case; the triangle-free case remains open
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algorithms-optimization Aug 11, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.
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analysis Apr 25, 2026 Significance 10/100 Registry: unreviewed
Prior state unknown → proved
Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference
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number-theory Jul 13, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
If g 3 ( n ) g_3(n) g 3 ( n ) is the largest size of A ⊆ [ 1 , n ] A \subseteq [1,n] A ⊆ [ 1 , n ] with fewer than three representations of every product a 1 a 2 a_1 a_2 a 1 a 2 , does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.
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algebra May 21, 2026 Significance 10/100 Registry: lean verified
Prior state unknown → proved
For a pure O-sequence h = ( h 0 , … , h e ) h = (h_0, \dots, h_e) h = ( h 0 , … , h e ) of codimension three and type two, is h i 2 ≥ h i − 1 h i + 1 h_i^2 \ge h_{i-1} h_{i+1} h i 2 ≥ h i − 1 h i + 1 for every interior index i i i ? The stated monomial case is proved; the broader level-Hilbert-function case remains open.
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