Question 50
Given a positive integer $n = \prod_{i=1}^{s} p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\sum_{i=1}^{s} \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12) = \lambda(2^2 \cdot 3^1) = (-1)^{2+1} = -1$).
Prove the following two claims:
i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$;
ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$.
(Romania) Dan Schwarz
Prove the following two claims:
i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$;
ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$.
(Romania) Dan Schwarz