MathDB

Problem 1

Part of 1998 VJIMC

Problems(2)

no primes in arithmetic seq of fixed length

Source: VJIMC 1998 1.1

8/24/2021
Let aa and dd be two positive integers. Prove that there exists a constant KK such that every set of KK consecutive elements of the arithmetic progression {a+nd}n=1∞\{a+nd\}_{n=1}^\infty contains at least one number which is not prime.
number theoryArithmetic Progression
eigenspace complement is invariant under linear operator

Source: VJIMC 1998 2.1

8/2/2021
Let HH be a complex Hilbert space. Let T:H→HT:H\to H be a bounded linear operator such that ∣(Tx,x)∣≤∥x∥2|(Tx,x)|\le\lVert x\rVert^2 for each x∈Hx\in H. Assume that μ∈C\mu\in\mathbb C, ∣μ∣=1|\mu|=1, is an eigenvalue with the corresponding eigenspace E={ϕ∈H:Tϕ=μϕ}E=\{\phi\in H:T\phi=\mu\phi\}. Prove that the orthogonal complement E⊥={x∈H:∀ϕ∈E:(x,ϕ)=0}E^\perp=\{x\in H:\forall\phi\in E:(x,\phi)=0\} of EE is TT-invariant, i.e., T(E⊥)⊆E⊥T(E^\perp)\subseteq E^\perp.
linear algberavector