MathDB
Golden set

Source: Iran 2004

September 18, 2004
number theoryprime numberscombinatorics proposedcombinatorics

Problem Statement

We say m∘nm \circ n for natural m,n ⟺\Longleftrightarrow nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say m∙nm \bullet n if and only if m,nm,n doesn't have the relation ∘\circ We say A⊂NA \subset \mathbb{N} is golden ⟺\Longleftrightarrow ∀U,V⊂A\forall U,V \subset A that are finite and arenot empty and U∩V=∅U \cap V = \emptyset,There exist z∈Az \in A that ∀x∈U,y∈V\forall x \in U,y \in V we have z∘x,z∙yz \circ x ,z \bullet y Suppose P\mathbb{P} is set of prime numbers.Prove if P=P1∪...∪Pk\mathbb{P}=P_1 \cup ... \cup P_k and Pi∩Pj=∅P_i \cap P_j = \emptyset then one of P1,...,PkP_1,...,P_k is golden.
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