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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2$\begingroup$ A counterexample is $N=2$. $\endgroup$– Philipp LampeCommented Aug 3, 2018 at 14:09
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3$\begingroup$ It holds whenever the summatory Liouville function $L(N-1)\le0$. Too bad Pólya's conjecture is false. $\endgroup$– Emil JeřábekCommented Aug 3, 2018 at 14:19
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2$\begingroup$ I guess this means the smallest counterexample is N= 906150257+1. $\endgroup$– kodluCommented Aug 3, 2018 at 14:38
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2$\begingroup$ @kodlu The condition I gave is only sufficient, not necessary. Since the statement follows from Goldbach's conjecture, it is likely true. In particular, I'm pretty sure it was verified for such small(ish) numbers. $\endgroup$– Emil JeřábekCommented Aug 3, 2018 at 16:35
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2$\begingroup$ Couple of quick observations - 1) we can assume $\lambda(N)=-1$ (or else $N=N/2+N/2$ will do) and 2) it would be enough to know (for large $N$) that $2p$ (for large $p$) is the sum of $a+b$ where $\lambda(a)=\lambda(b)=1$, since $N=aN/2p+bN/2p$. $\endgroup$– Thomas BloomCommented Aug 4, 2018 at 10:30
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1 Answer
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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.
answered Aug 5, 2018 at 13:54
Jan-Christoph Schlage-PuchtaJan-Christoph Schlage-Puchta
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1$\begingroup$ I could not find the results of Pintz you are referring to. $\endgroup$– PabloCommented Aug 5, 2018 at 16:03
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5$\begingroup$ Actually there is (surprisingly) a significant jump in difficulty when trying to use the GPY sieve to get Goldbach-type results rather than bounded gap type results, in particular there is a parity obstruction: terrytao.wordpress.com/2014/07/09/… . There is still a slight chance that current methods are strong enough to resolve this problem, but it seems nontrivial to me. $\endgroup$ Commented Aug 5, 2018 at 22:20
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$\begingroup$ @Pablo: Pintz gave talks about it. I know that he has a stack of unpublished manuscripts, so possibly his one is among them. The role of the parity problem is not really clear to me. $\endgroup$ Commented Aug 7, 2018 at 15:41
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1$\begingroup$ The statement $\omega(n)=\omega(n+1)=k$, for every $k\geq 3$ is Theorem 2 in "Small gaps between almost primes, the parity problem and some conjectures of Erdös on consecutive integers" Goldston, Graham, Pintz, and Yildirim arxiv.org/abs/0803.2636 $\endgroup$ Commented Aug 12, 2018 at 16:08