  Let $\lambda(n)$ and $\mu(n)$ denote the Liouville function and the M\"obius
function, respectively. In this study, relationships between the values of
$\lambda(n)$ and $\lambda(n+h)$ up to $n\leq10^8$ for $1\leq h\leq1,000$ are
explored. Chowla's conjecture predicts that the conditional expectation of
$\lambda(n+h)$ given $\lambda(n)=1$ for $1\leq n\leq X$ converges to the
conditional expectation of $\lambda(n+h)$ given $\lambda(n)=-1$ for $1\leq
n\leq X$ as $X\rightarrow\infty$. However, for finite $X$, these conditional
expectations are different. The observed difference, together with the
significant difference in $\chi^2$ tests of independence, reveals hidden
additive properties among the values of the Liouville function. Similarly, such
additive structures for $\mu(n)$ for square-free $n$'s are identified. These
findings pave the way for developing possible, and hopefully efficient,
additive algorithms for these functions. The potential existence of fast,
additive algorithms for $\lambda(n)$ and $\mu(n)$ may eventually provide
scientific evidence supporting the belief that prime factorization of large
integers should not be too difficult. For $1\leq h\leq1,000$, the study also
tested the convergence speeds of Chowla's conjecture and found no relation on
$h$.
