On an Erdős problem concerning finite unions of multiples: counterexamples and positive results
Year: 2026
Type: preprint
Abstract: Take a finite set A of positive integers and let F_A(x) count the integers in [1, x] divisible by at least one element of A. Erdős asked a simple question: is F_A(m)/m < 2·F_A(n)/n whenever m > n ≥ max A? This is Erdős Problem #488, and the answer is no. In fact, no constant can replace 2. The counterexamples come from smooth numbers. Fix y, let q be the next prime after y, choose an integer c with y ≤ c < q, and let A be the y-smooth integers in [Y, cY]. Up to n = cY, the multiples of A produce nothing new: F_A(n) = |A| exactly. Far beyond n, every multiple au with u free of small primes is counted, and the density of the multiples becomes exactly ∏_{p≤y}(1 − 1/p) · Σ 1/d, summed over y-smooth d ≥ Y. Overlap at small scales, freedom at large ones: that is the entire counterexample. An averaging argument then shows that the supremum of the ratio (F_A(m)/m)/(F_A(n)/n) is at least ∏_{p≤y}(1 − 1/p)·(c − 1)/log c, which grows without bound. For y = c = 100 and Y = 10^87, the ratio exceeds 2.27. Two independently written programs certify this, one of them in exact integer arithmetic. Yet the inequality is far from hopeless. I prove it when A has at most three generators after removing redundant elements, and when no three elements of A have pairwise least common multiples at most n. An exhaustive search in exact arithmetic proves the sharp bound F_A(m)/m ≤ F_A(n)/(⌊n/2⌋ + 1) for every A with max A ≤ n ≤ 93 and every m > n.
Author: Girma, Samuel
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225348508