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Semiprimes (pq-numbers) guarantee that if p and q are prime numbers the only divisors of the result of p*q are p and q.

My question is: Does this hold true for p*q*r as well?

For example 3*5*7=105. Are there any three numbers other than 3, 5 and 7 so that x*y*z=105?

Of course if there are non, the follow up question is, what about the other prime-multiplications like p*q*r*s?

I have this question because I came across an article about sending messages to space and the Arecibo message used a semiprime to transmit information about the layout of the message. If we could do the same with three numbers we could make a 3d layout.

The cardinality of 1,679 was chosen because it is a semiprime (the product of two prime numbers), to be arranged rectangularly as 73 rows by 23 columns. The alternative arrangement, 23 rows by 73 columns, produces jumbled nonsense (as do all other X/Y formats). The message forms the image shown on the right, or its inverse, when translated into graphics, characters and spaces.[5]

from the Wikipedia page

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  • $\begingroup$ I am not sure whether I have understood the intent of the question, but it seems that you want to know whether a number which is a product of $n$ distinct primes can essentially only be written as a product of $n$ factors in one way. If so, the answer is yes, if we do not allow $1$ as a factor. $\endgroup$ – Peter Jul 11 '17 at 21:05
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    $\begingroup$ A 3D layout is much more complex because $3\times 5\times 7 =5\times 3 \times 7= 7\times 3 \times 5\ldots$ there are 6 possibilities, not only 2 like in 2D and the aliens would find hard time reconstructing the message $\endgroup$ – Raffaele Jul 12 '17 at 15:29
  • $\begingroup$ @Raffaele I didn't even think about that as I wrote my answer. I think you should add an answer, expanding on your comment. $\endgroup$ – Robert Soupe Jul 13 '17 at 19:02
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If $p$ and $q$ are distinct prime numbers, then $pq$ actually has four divisors in $\mathbb Z^+$: $1, p, q, pq$. What you want to know about are what we might call "nontrivial" divisors, divisors other than 1 and the number itself. In which case $pq$ does indeed have two nontrivial divisors: $p$ and $q$.

If $p$, $q$ and $r$ are distinct prime numbers, then $pqr$ does indeed have $p$, $q$ and $r$ among its nontrivial divisors. But it also has $pq$, $pr$ and $qr$ among its nontrivial divisors.

To use your example of 105: its nontrivial divisors are 3, 5, 7, 15, 21, 35. So if we sent the aliens a 3 by 5 by 7 cuboid array, they might misunderstand it as a 3 by 35 rectangular array, or a 5 by 21, or 7 by 15.

Even with a $p$ by $q$ array, there is the danger they might misunderstand it. With more possible ways to interpret the message, the potential for misunderstanding increases, along with the potential for star wars.

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