For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.
Problem
Source: Chinese TST 2 2013 Day 2 Q1
Tags: symmetry, number theory proposed, number theory
29.03.2013 10:04
Quite easy this one. Just take $x=1,a_1+2,a_2+2,\cdots,a_n+2$, and that's finished.
29.03.2013 10:31
Note that for arbitrary integers $x_1,x_2,...,x_n$ we have $\Omega (p(x_1)p(x_2)...p(x_n))$ is even due to the fact that $\Omega(ab) = \Omega(a) + \Omega(b)$. Now, remark that: \[p(1) \cdot \prod_{i=1}^n p(a_i + 2) = 2^n \cdot \prod_{i=1}^n (a_i + 1)^2 \prod_{1 \le i < j \le n} (a_i + a_j + 1)^2 \] However, $\Omega$ of this is congruent to $n$ modulo $2$. It immediately follows that $n$ is even. To motivate this solution, it is clear that to solve this problem we should multiply $p(x_i)$ together for various $x_i$. We want to get a lot of squares of expressions. Thus plugging in some sort of $a_i + k$ into the expressions is convenient due to the high level of symmetry present. This gets some $2a_i + k$ terms, so $2|k$ would allow us to pull out $n$ powers of $2$'s which is extremely useful. Then plug in $k/2$ to get only a $2^n$ remaining. Letting $k=2$ yields the above solution.
19.08.2021 18:55
Similar to @dinoboy's Since $\Omega(ab) = \Omega(a) + \Omega(b)$ it follows that $\Omega (p(x_1)p(x_2)...p(x_n))$ is even. Clearly, $p(1) \cdot \prod_{i=1}^n p(a_i + 2) = 2^n \cdot \prod_{i=1}^n (a_i + 1)^2 \prod_{1 \le i < j \le n} (a_i + a_j + 1)^2 $ which implies $n $ is even.
23.06.2022 20:20
24.07.2023 08:51
Sketch: It is common knowledge that $\Omega(ab)$ is completely additive. Hence, $2|\Omega (p(x_1)p(x_2)...p(x_n))$ and the solution follows.
04.03.2024 18:16
My question: Does there exist pairwise distinct positive integers $a_1,a_2,...,a_n$ satisfying the requirement?