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Give an element of $ \mathbb{Z}[\sqrt{-17}] $ that is a product of two irreducibles and also a product of three irreducibles.

My thoughts so far:

Using the multiplicative norm $ N(a + b\sqrt{-17}) = a^2 + 17 b^2 $, we see that the units are precisely 1, -1. I can also see that there are no elements of norm $ 2,3,5,6,7,8,10,11,12,13,14,15... $. So if an element has norm 4 or 9 for example, then it is irreducible.

I don't really know where to go from here.

Any help appreciated. Thanks

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    $18=2\cdot3\cdot3=(1-\sqrt{-17})(1+\sqrt{-17})$2011-05-06
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    Now that you have the reputation for it, don't forget to vote up answers you find useful and questions you find interesting (as a participant). Also, don't do it just yet (I'll explain why in a second), but don't forget to eventually *accept* an answer to your questions once you are satisfied. You should accept whatever answer you found most intersting/helpful, etc, by clicking on the checkmark you will see on the left margin, right under the links to vote up and down (you don't have the reputation to vote down yet). (cont...)2011-05-06
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    (cont) You may want to wait before accepting answers to your recent questions because questions without an accepted answer tend to attract a *bit* more attention, and you may yet receive other answers that prove to be better/more interesting/more informative/ etc. Waiting a day or so is not amiss, but you'll want to *eventually* accept an answer to each of your questions once you are satisfied.2011-05-06

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