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Forehead slapper #1

  1. Apr 4, 2004 #1
    As a little diversion I thought i'd post this question which I call a forehead-slapper because that's what you'll likely do when you see the answer. You won't need more than high school maths to solve it.


    Show that if x and [tex]x^2+8[/tex] are primes then so is [tex]x^3-8[/tex]
     
  2. jcsd
  3. Apr 5, 2004 #2
    x = 3?

    cookiemonster
     
  4. Apr 5, 2004 #3

    Zurtex

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    Well [itex]x^3 - 8 = (x - 2)(x^2 + 2x + 4) [/itex] meaning that this can only be a prime when [itex]x = 3[/itex].

    Edit: I think I have proved the rest of it I'll let others have a go.
     
    Last edited: Apr 5, 2004
  5. Apr 5, 2004 #4
    There's a rest of it?

    cookiemonster
     
  6. Apr 5, 2004 #5

    Janitor

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    Given the Zurtex factorization, it seems pretty obvious that for x=4, 5, 6, ..., x^3-8 has to be composite, so I'm with Cookiemonster.
     
  7. Apr 5, 2004 #6
    Zurtex is exactly right. That was exactly the solution I had in mind. There is another one though I just realized, that doesn't involve having to factor a cubic.

    Note that any prime p except 3 is equal to 3k+1 or 3k-1 for some integer k. Then p^2 + 8 = 9k^2 ± 6k + 9, which is divisible by 3. 3^2+8=17, which is prime and also 3^3-8=19 is prime.
     
  8. Apr 5, 2004 #7

    Zurtex

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    Yes, the rest of it was obviously to prove that [itex]x[/itex] and [itex]x^2 + 8[/itex] could never both be prime. My proof was a little bit more complex that as it is early in the morning and I can't think simple maths yet :rolleyes:
     
  9. Apr 5, 2004 #8

    HallsofIvy

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    The point of cookiemonster's post was that x= 3 is a prime number and that x2[/sup+8= 9+8= 17 is a prime number but x3= 27-8= 21= 3*7 is NOT.

    You can't prove your statement: it's not true.

    What Zurtex showed with "[itex]x^3 - 8 = (x - 2)(x^2 + 2x + 4) [/itex]" was that x3- 8 cannot be prime unless x= 3. That is essentially a converse of your original statement.
     
  10. Apr 5, 2004 #9

    matt grime

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    27-8 = 19 <filler space>
     
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