Problems

Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?

Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives κ ≤ 1 + √1 - ℜ⟨ E_1…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be - in particular, can the chromatic number be infinite? Yes: there is such a set, no three…

For a connected graph G , let t= tree( G ) (order of a largest induced tree), A= average eccentricity, and L= maximum independence number of a neighbourhood. Then ⌈ (A+L)/3 ⌉ ≤ t. (The evenly-divided reading of the conjecture holds; a…

If each integer has at most r representations m = pa with p prime and a ∈ A ⊆ [1, N], what is the best upper bound for ∑_a ∈ A 1/a? The candidate proof gives the matching order Θ_r(log N / loglog N).

For the least cutoff c(n) after which every k occurs as the number of homothetic cubes in a decomposition of the unit n-cube, is c(n) ≫ n^n? The Lean proof shows c(n) = o(n^n) along odd dimensions.

If n planar points have no four concyclic, must some point determine (1 - o(1))n distinct distances? Failing that, can one always force more than (1/3 + c)n?

Determine the Shannon capacities of odd cycles beyond C_5, or improve the best explicit bounds. Lovasz's theta function settled C_5 in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model…

Let a,b,c>1 be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form a^k b^l c^m (k,l,m≥ 0), none dividing another?

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Is the number of nonnesting permutations of 1,1,…,n,n avoiding both 1132 and 3312 equal to 3^n - 3 · 2^n-1 + 1 for every n ≥ 1?

VibeMathed records no statement for this problem. See formal-conjectures PR #4668 - Mark WOWII Graph Conjecture 217 solved for the original.

Let F(N) be the maximal size of A⊆1,…,N such that no a∈ A divides the sum of any nonempty subset of A∖a. Estimate F(N). The lower bound F(N)≫ N^1/5 is classical, from constructions of Erdős and Csaba, and every non-dividing set is…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Erdős conjectured, in over a dozen papers spanning 1976 to 1997 and with a $1000 prize attached, that every finite Sidon set extends to a perfect difference set modulo p^2+p+1 for some prime p. Alexeev and Mixon establish that 1,2,4,8 is a…

Jaeger conjectured that every bridgeless cubic graph G admits a Petersen coloring: a map φcolon E(G)→ E(P) into the edges of the Petersen graph P such that, for every vertex v of G, the three edges at v are sent to three edges meeting at a…

In the half-collinear single-minus graviton recursion of Guevara, Lupsasca, Skinner, Strominger and Weil, the multipoint vertex weights depend on global cut tests, which blocks a direct matrix-tree formula outside a restricted decay…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Let X = (X_1,…,X_n) be a centered Gaussian vector, not necessarily nondegenerate. Then, for every α_1,…,α_n > 0, E[∏_i=1^n |X_i|^α_i] ≥ ∏_i=1^n E[|X_i|^α_i]. Moreover, if Var(X_i) > 0 for every i, then equality holds if and only if…