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Let $m,n\in \mathbb{Z}$, what is the notation usually used to say that $m,n$ have the same prime factors, i.e. $m=p_1^{m_1}p_2^{m_2}\cdots p_2^{m_r}$, $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$ for some primes $p_1,p_2\cdots,p_r$.

I think about Radical(n)=Radical(m), but maybe there is a better notation?

Bests.

  • It is possible to write $m,n\in \langle p_1p_2 \dots p_r \rangle$ - both are elements of the ideal generated by the product of the primes. But that depends on the context (whether ideals are part of it or not) - you could write the same anyway. – Mark Bennet Aug 21 '14 at 16:39
  • @user31009: Is my answer not what you are seeking after? – Yes Aug 23 '14 at 12:29

1 Answers1

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The radical operator may suit your need. If $m$ is an integer such that $m = \Pi_{1}^{r}p_{j}^{m_j}$ for some integer $r > 0$ and some integers $m_{1}, \dots, m_{r} \geq 0$ and some primes $p_{1}, \dots, p_{r}$, the radical of $m$ is defined to be $\Pi_{1}^{r}p_{j}$ and is denoted by $rad(m)$. Then two integers $m, n$ have the same prime divisors if and only if $$rad(m) = rad(n).$$

Yes
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