Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
21 problems
Let S(x) count ordered pairs (a,b) with a+b ≤ x and σ(a)+σ(b) = σ(a+b). Erdos asked whether S(x) ~ cx. The opposite extreme holds: for every R > 0, S(x)/(x(log x)^R) → ∞, so the count beats every fixed logarithmic scale.
Ross introduced S-perfect numbers, integers expressible as 1 + ∑ λ_j d_j over their proper divisors with coefficients in S, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Erdős and Graham asked whether binomnk with 1 ≤ k ≤ n/2 must always have a divisor ≤ n that is close to n, meaning bigger than a fixed constant times n. Settled in both directions: true when k is large enough as a function of n, but false…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For an odd prime p, do Sun's normalized trigonometric permanents satisfy s_p < 0 ⇔ p ≡ 5 pmod12 and s'_p < 0 ⇔ p ≡ 7 pmod 8? Exact computation at p = 29 refutes both sign laws.
If Sidon sets A, B ⊆ 1, …, N satisfy (A-A) ∩ (B-B) = 0, must binom|A|2 + binom|B|2 ≤ binomf(N)2 + O(1), where f(N) is the largest Sidon-set size in [N] - and can the bound be improved by a fixed proportion when |A| = |B|?
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
An n-divisor set contains a multiple of every integer from 1 to n. Umans and Wang proposed, as the arithmetic-progression form of their Strong (α,β)-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude,…
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…
For A ⊆ F_p let A^* = (A+A) ∪ (AA). Sárközy conjectured that for all large primes, every set of size at least c√p has A^* = F_p-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together…
For the Erdős–Pomerance functions F(n) and h_P(n) counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to n, the paper proves F(n) ≥ h_P(n) ≥ nexp((log 2/2 - o(1))log…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Erdos and Graham asked whether a positive-density subset of 1,…,N can avoid having any two distinct elements a,b whose unit fractions average to a unit fraction. It can: there is a constant c>0 such that for all large N some A ⊆ 1,…,N of…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
A monic prime P of F_q[T] is a c-Wieferich prime if ρ_P(1) ≡ 1 bmod P^2 for the Carlitz module ρ. On limited data and proofs in degrees 2 and 3, Thakur suggested in 2015 that in odd characteristic every c-Wieferich prime has degree…