Is it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ?
Here $\lambda$ is the Liouville function.
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Sign up to join this communityIs it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ?
Here $\lambda$ is the Liouville function.
In "The equation $\omega(n)=\omega(n+1)$" (Mathematika 50, 99-101, 2003) it was shown that the equation $\omega(n)=\omega(n+1)$ has infinitely many solutions. Pintz has a series of results on consecutive integers, one of which is that for every $k>k_0$ we have that the equation $\omega(n)=\omega(n+1)=k$ has infinitely many solutions, where $k_0$ is some small integer (may be 5). Unfortunately the closest reference I could find is "Small gaps between products of two primes" by Goldston, Graham, Pintz, and Yildirim. However, I guess a similar application of the GPY-sieve should give that for all sufficiently large $N$ and $k>k_0$ we have that $n+m=N$ with $\omega(n)=\omega(m)=k$ has solutions. Going from $\omega$ to $\Omega$ should not pose any problems, so even a stronger statement should be true and provable with today's methods.