MathDB

Given a positive integer

\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}

, we write

\Omega(n)

for the total number

\displaystyle \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\cdot3^1)=(-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

Problem Statement