exist infinitely many positive integers $n$ such $\omega (n)+\omega (n+1)\equiv 0\pmod 3$

Question

For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$. Prove that:there exist infinitely many positive integers $n$ satisfying $$\omega (n)+\omega (n+1)\equiv 0\pmod 3$$ Positive integers $n$ satisfying $\omega(n)<\omega(n+1)<\omega(n+2)$

Answer 1

This proof is based on some of the ideas used in the answer by reuns. Assume there are a finite # of $n$ where

$$\omega(n) + \omega(n+1) \equiv 0 \pmod 3 \tag{1}\label{eq1}$$

with $n_0$ being the largest value. Let $n_1 \gt n_0$ be an odd integer such that

$$\omega(2n_1) \equiv 0 \pmod 3 \tag{2}\label{eq2}$$

such as with $n_1 = pq$ for $2$ distinct odd primes $p$ and $q$. Next, let

$$\omega(2n_1 + 1) \equiv a \pmod 3 \tag{3}\label{eq3}$$ $$\omega(2n_1 + 2) \equiv b \pmod 3 \tag{4}\label{eq4}$$ $$\omega(2n_1 + 3) \equiv c \pmod 3 \tag{5}\label{eq5}$$ $$\omega(2n_1 + 4) \equiv d \pmod 3 \tag{6}\label{eq6}$$

The rest of this long & detailed proof will demonstrate that $a \equiv c \equiv 2 \pmod 3$ and $b \equiv d \equiv 0 \pmod 3$, i.e., the $\omega$ value congruences will always alternate between $0$ and $2$ only. To show this, first note that

$$\omega(2n_1) + \omega(2n_1 + 1) \equiv 0 + a \equiv a \not\equiv 0 \pmod 3 \tag{7}\label{eq7}$$ $$\omega(2n_1 + 1) + \omega(2n_1 + 2) \equiv a + b \not\equiv 0 \pmod 3 \tag{8}\label{eq8}$$

Since $n_1$ is odd, it would have one less distinct prime factor than $2n_1$ in \eqref{eq2}, so

$$\omega(n_1) \equiv 2 \pmod 3 \tag{9}\label{eq9}$$

As $2n_1 + 2 = 2\left(n_1 + 1\right)$, and $n_1 + 1$ is even, this means that the $\omega$ value of $n_1 + 1$ is the same as for $2n_1 + 2$ in \eqref{eq4}, so

$$\omega(n_1 + 1) \equiv b \pmod 3 \tag{10}\label{eq10}$$

Thus, since the sum of \eqref{eq9} and \eqref{eq10} can't be zero modulo $3$, this gives

$$\omega(n_1) + \omega(n_1 + 1) \equiv 2 + b \not\equiv 0 \pmod 3 \; \Rightarrow \; b \not\equiv 1 \pmod 3 \tag{11}\label{eq11}$$

Since $2n_1$ and $2n_1 + 2$ have a common factor of only $2$, this means their product has an $\omega$ value which is one less than their sum, i.e.,

$$\omega(4n_1^2 + 4n_1) = \omega(2n_1) + \omega(2n_1 + 2) - 1 \equiv b - 1 \pmod 3 \tag{12}\label{eq12}$$

Also, the square of $2n_1 + 1$ would have the same $\omega$ value, so

$$\omega(4n_1^2 + 4n_1 + 1) \equiv a \pmod 3 \tag{13}\label{eq13}$$

Adding \eqref{eq12} and \eqref{eq13} gives

$$\omega(4n_1^2 + 4n_1) + \omega(4n_1^2 + 4n_1 + 1) \equiv a + b - 1 \not\equiv 0 \pmod 3 \tag{14}\label{eq14}$$

\eqref{eq7} only allows $a \equiv 1,2 \pmod 3$. With $a \equiv 1 \pmod 3$, \eqref{eq8} and \eqref{eq11} only allow $b \equiv 0 \pmod 3$. However, this fails in \eqref{eq14}. With $a \equiv 2 \pmod 3$, \eqref{eq8} and \eqref{eq11} both give that $b \not\equiv 1 \pmod 3$ while \eqref{eq14} gives $b \not\equiv 2 \pmod 3$, so $b \equiv 0 \pmod 3$ is the only value allowed, giving that $a \equiv 2 \pmod 3$ and $b \equiv 0 \pmod 3$, with adding \eqref{eq4} & \eqref{eq5} then showing $c \not\equiv 0 \pmod 3$. Since $2n_1 + 1$ and $2n_1 + 3$ are both odd & relatively prime, the $\omega$ of their product would be the sum of their $\omega$ values from \eqref{eq3} and \eqref{eq5}, so

$$\omega(4n_1^2 + 8n_1 + 3) \equiv a + c \equiv 2 + c \pmod 3 \tag{15}\label{eq15}$$

Also, the square of $2n_1 + 2$ would have the same $\omega$ value in \eqref{eq4} so

$$\omega(4n_1^2 + 8n_1 + 4) \equiv b \equiv 0 \pmod 3 \tag{16}\label{eq16}$$

Adding \eqref{eq15} and \eqref{eq16} gives

$$\omega(4n_1^2 + 8n_1 + 3) + \omega(4n_1^2 + 8n_1 + 4) \equiv 2 + c \not\equiv 0 \pmod 3 \tag{17}\label{eq17}$$

Thus, $c \not\equiv 1 \pmod 3$ and as we determined earlier that $c \not\equiv 0 \pmod 3$, this means that $c \equiv 2 \pmod 3$. Next, checking the sum of \eqref{eq5} and \eqref{eq6} gives

$$\omega(2n_1 + 3) + \omega(2n_1 + 4) \equiv c + d \equiv 2 + d \not\equiv 0 \pmod 3 \tag{18}\label{eq18}$$

Repeating the basic procedures of above of multiplying $2n_1 + 2$ and $2n_1 + 4$, plus squaring $2n_1 + 3$, and using their $\omega$ values in \eqref{eq4} to \eqref{eq6} gives

$$\omega(4n_1^2 + 10n_1 + 8) + \omega(4n_1^2 + 10n_1 + 9) \equiv b + d - 1 + c \equiv d + 1 \not\equiv 0 \pmod 3 \tag{19}\label{eq19}$$

Thus, $d \not\equiv 2 \pmod 3$, while \eqref{eq18} gives that $d \not\equiv 1 \pmod 3$, so $d \equiv 0 \pmod 3$. However, $2n_1 + 4 = 2(n_1 + 2)$, so this mean the original $n_1$ can be replaced by $n_1 + 2$ and the above procedure repeated over and over again, giving that the $\omega$ values are always alternating between being congruent to $0$ and $2$ modulo $3$, as stated originally. Thus, the $\omega$ value will never by congruent to $1$ modulo $3$, which is of course impossible such as for cases where you have a prime number. This means the original assumption that there is only a finite # of $n$ which satisfy \eqref{eq1} must be false, i.e., there actually are an infinite # of them.

Answer 2

With $\Omega$ instead of $\omega$ I have a solution.

For any $n$ then $a = \Omega(2n) \bmod 3,b=\Omega(2n+1)\bmod 3,c=\Omega(2n+2)\bmod 3$, $\Omega((2n+1)^2) = 2b\bmod 3, \Omega((2n+1)^2-1) = a+c\bmod 3,\Omega(n)+\Omega(n+1) = a+c-2 \bmod 3$

Assume we chose $n$ such that $a=2\bmod 3$

The claim $\Omega(l)+\Omega(l+1) = 0 \bmod 3$ for some $l \in \{ 2n,2n+1, (2n+1)^2-1,n\}$ would follow from

$2+b=0\bmod 3$ or $b+c=0\bmod 3$ or $2+c+2b=0\bmod 3$ or $2+c-2=0\bmod 3$.

The only fail is when $c=2\bmod 3$.

But if $c=2\bmod 3$ we can redo it replacing $n$ by $n+1$, finding that for the claim to fail we need $\Omega(2(n+m)) = 2\bmod 3$ for every $m\ge 0$, a contradiction.