Problems

Can there be a finite covering system of the integers with distinct moduli, all of which are odd and greater than 1?

Let A ⊂ N be infinite with no distinct a, b, c ∈ A such that a | (b + c) with b, c > a. Can |A ∩ [1, N]|/√N have positive lower limit? Must every such A fall below N^1-c infinitely often?

Let A(x) count n ≤ x such that every prime p | n has a divisor d > 1 of n with d ≡ 1 pmod p. Erdos asked whether A(x)/x = exp(-(c+o(1))√log xloglog x). It does, with c = 1/(2√log 2).

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

Let D_q(n) be the largest possible least degree of a polynomial omitted by a non-covering family of n distinct-modulus congruence classes in F_q[x]. What is its asymptotic size? The answer is D_q(n) = n/q-1 + O_q(1).

Does there exist A=a_1<a_2<…⊂ N which is a minimal basis of order 2 (every large integer is the sum of 2 elements from A, and no proper subset of A has this property) such that lim_k→ ∞a_k/k^2=c for some c≠ 0? A claimed construction gives…

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.

Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)…(n+k))>n^1-ε is at least 1-η, where P(m) is the greatest prime divisor of m? A short argument via the Matomäki-Radziwiłł theorem establishes the…

If A(x) counts integers satisfying the Sylow divisor condition, determine the constant c in A(x)/x = exp(-(c + o(1)) √log x loglog x). The claimed exact value is c = 1/(2√log 2).

Does there exist an integer polynomial f of degree at least two and a set A ⊆ Z such that every integer has a unique representation n = a + f(k)? A manuscript claims the thirteenth powers admit a tiling complement.

For |A| = n, how small can the cofactor set Q(A) = a / gcd(a,b) : a, b ∈ A be? The answer is h(n) = n^1/2 + o(1): a new upper bound h(n) ≤ n^1/2 exp(O(√log n)) matches the classical lower bound.

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

Bounds the weighted sum ∑ 1/(a log a) taken over primitive sets of integers (sets where no element divides another).

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…

What is the largest A⊆1,…,N such that all subset sums ∑_n∈ S1/n (over S⊆ A) are distinct?

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

For every finite set A⊂ Z with |A|≥ 2, define C(A)=log(|A+A|/|A|)/log(|A-A|/|A|). Determine the largest possible value of C(A), equivalently the least universal exponent c such that |A+A|/|A| ≤ (|A-A|/|A|)^c for every such set A. The…

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.