Let $A=\{n_1<\cdots <n_r\}$ be a finite set of positive integers. What is the maximum density of integers covered by a suitable choice of congruences $a_i(\mathrm{mod}{n}_i)$?Is the minimum density achieved when all the $a_i$ are equal?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Fix moduli (A={n_1<\cdots<n_r}) and choose residue classes [ C_i={m\in\mathbb Z:m\equiv a_i\pmod{n_i}}. ] Let (U(a_1,\dots,a_r)=\bigcup_{i=1}^r C_i). If (L=\mathrm{lcm}(n_1,\dots,n_r)), then (U) is periodic mod (L), hence it has a (natural) density and $ d(U(a_1,\dots,a_r))=\frac{|{0\le x<L:\exists i,\ x\equiv a_i\ (\m…

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

2 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.