clopen.solutions

Open problems

1131 registered problems.

OP-00001 Open
Goldbach's conjecture
NT · additive prime number theory · Landau's problems
OP-00002 Open
The twin prime conjecture
NT · Landau's problems · prime gaps
OP-00003 Open
Legendre's conjecture
NT · Landau's problems · prime gaps
OP-00004 Open
Primes of the form n² + 1
NT · Landau's problems · primes represented by polynomials
OP-00005 Open
The odd perfect number conjecture
NT · perfect numbers · sum of divisors
OP-00006 Open
Lehmer's totient problem
NT · Euler's totient function
OP-00007 Open
The Kummer–Vandiver conjecture
NT · class numbers · cyclotomic fields
OP-00008 Open
Pillai's conjecture on axⁿ − byᵐ = c
NT · exponential Diophantine equations · perfect powers
OP-00009 Open
The Feit–Thompson conjecture on primes
NT · divisibility
OP-00010 Open
The Lander–Parkin–Selfridge conjecture
NT · Diophantine equations · equal sums of like powers
OP-00011 Open
A 3 × 3 magic square of distinct squares
NT · Diophantine equations · magic squares
OP-00012 Open
Schanuel's conjecture
NT · exponential function · transcendental numbers
OP-00013 Open
The four exponentials conjecture
NT · exponential function · transcendental numbers
OP-00014 Open
Irrationality of the Euler–Mascheroni constant
NT · Euler–Mascheroni constant · irrationality
OP-00015 Solution claimed
Convergence of the Flint Hills series
CA · irrationality measure · series convergence
OP-00016 Solution claimed
Falconer's distance set conjecture
CA · distance sets · geometric measure theory · Hausdorff dimension
OP-00017 Partially resolved
The Marcus–de Oliveira determinantal conjecture
RA · determinants · normal matrices
OP-00018 Open
The 1/3–2/3 conjecture for finite posets
CO · linear extensions · partially ordered sets · sorting
OP-00019 Open
The prime power conjecture for finite projective planes
CO · Erdős problems · finite projective planes
OP-00020 Solution claimed
The Gaussian moat problem
NT · Erdős problems · Gaussian primes
OP-00021 Open
Interior of the spectrum of the boat–star tiling Laplacian
SP · aperiodic order · Cantor spectrum · graph Laplacian · Penrose tiling
OP-00022 Open
Intervals of spectrum at the edges of the boat–star tiling spectrum
SP · aperiodic order · graph Laplacian · Penrose tiling · spectral edges
OP-00023 Open
Closed form for the top of the boat–star tiling spectrum
SP · aperiodic order · graph Laplacian · Penrose tiling · spectral edges
OP-00024 Open
Tilings without locally supported eigenfunctions in every MLD class
SP · aperiodic order · graph Laplacian · locally supported eigenfunctions · mutual local derivability
OP-00025 Open
Finitely many jumps in the integrated density of states of a linearly repetitive tiling
SP · aperiodic order · graph Laplacian · integrated density of states · linear repetitivity · locally supported eigenfunctions
OP-00026 Open
Hölder continuity of the boat–star integrated density of states at the spectral edges
SP · aperiodic order · Hölder continuity · integrated density of states · Penrose tiling · spectral edges
OP-00027 Open
Interior, edge intervals and edge regularity for the Robinson triangle tiling Laplacian
SP · Cantor spectrum · graph Laplacian · integrated density of states · Penrose tiling · Robinson triangles
OP-00028 Open
Spectral gaps of the Robinson triangle tiling Laplacian
SP · Cantor spectrum · graph Laplacian · Penrose tiling · Robinson triangles · spectral gap
OP-00029 Solution claimed
Global synchronization of random 3-regular graphs
PR · global synchronization · Kuramoto model · random regular graphs · Randomstrasse101
OP-00030 Open
Sub-exponential detection in the hard regime of multi-frequency synchronization
ST · group synchronization · Randomstrasse101 · spiked random matrices · statistical-to-computational gaps
OP-00031 Open
The largest discrepancy of square ±1 matrices relative to √n
CO · discrepancy theory · Randomstrasse101 · sign matrices · Spencer's theorem
OP-00032 Open
The Hadamard conjecture
CO · combinatorial designs · Hadamard matrices · Randomstrasse101
OP-00033 Partially resolved
Fast mixing of Glauber dynamics for the SK model up to β = 1
PR · Glauber dynamics · mixing time · Randomstrasse101 · Sherrington–Kirkpatrick model · spin glasses
OP-00034 Open
A type-2 inequality for injective norms of Gaussian sums of tensors
PR · injective norm · matrix concentration inequalities · Randomstrasse101 · random tensors
OP-00035 Partially resolved
The Lovász number of random circulant graphs
CO · circulant graphs · Lovász theta function · random graphs · Randomstrasse101
OP-00036 Partially resolved
Vinzant's conjecture on 4M − 5 phase retrieval measurements
FA · frames · injectivity · phase retrieval · Randomstrasse101
OP-00037 Open
The Balan–Wang conjecture on the stability of real phase retrieval
FA · frames · phase retrieval · Randomstrasse101 · stability
OP-00038 Partially resolved
A degree-4 Sum-of-Squares proof that ℂ⁶ has no seven mutually unbiased bases
OC · mutually unbiased bases · Randomstrasse101 · semidefinite programming · Sum-of-Squares
OP-00039 Solution claimed
Zauner's conjecture
FA · equiangular tight frames · quantum information · Randomstrasse101 · SIC-POVM
OP-00040 Open
The clique number of Paley graphs
NT · clique number · Paley graph · pseudorandom graphs · Randomstrasse101
OP-00041 Open
The Lovász number of the 1-localization of the Paley graph
CO · clique number · Lovász theta function · Paley graph · Randomstrasse101
OP-00042 Open
A Lovász-number bound for the 2-localization of the Paley graph
CO · clique number · Lovász theta function · Paley graph · Randomstrasse101
OP-00043 Open
A degree-4 Sum-of-Squares bound for the Paley clique number
OC · clique number · Paley graph · Randomstrasse101 · Sum-of-Squares
OP-00044 Open
The Paley ETF restricted isometry conjecture
FA · compressed sensing · equiangular tight frames · Paley graph · Randomstrasse101 · restricted isometry property
OP-00045 Solution claimed
The Kannan–Lovász–Simonovits conjecture
PR · convex geometry · isoperimetry · KLS conjecture · log-concave measures · Randomstrasse101
OP-00046 Open
Minimum and maximum bisections of random regular graphs
PR · graph bisection · maximum cut · random regular graphs · Randomstrasse101 · spin glasses
OP-00047 Open
Directional convex hulls that are not semialgebraic
MG · directional convexity · Randomstrasse101 · rank-one convexity · semialgebraic sets
OP-00048 Open
When are directional convex hulls semialgebraic?
MG · directional convexity · Randomstrasse101 · rank-one convexity · semialgebraic sets
OP-00049 Open
Rigorous consequences of low-degree hardness
ST · hypothesis testing · low-degree polynomials · Randomstrasse101 · statistical-to-computational gaps
OP-00050 Open
A low-degree polynomial that finds cliques of size log₂ n in G(n, 1/2)
PR · clique number · low-degree polynomials · random graphs · Randomstrasse101