Problem Solving Set 19

21 July 2012

1. Let 0 < a < b. Prove that

∫_{a}^{b}(x^{2} + 1)e^{−x^{2}} dx ≥ e^{−a^{2}} − e^{−b^{2}}.

2. If n ∈ ℕ has the property that for all x ∈ ℤ there exists y ∈ ℤ such that x^{2} + y^{2} ≡ 1 (mod n), show that n divides