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Prove or disprove: Is 10^1,000 - 9 Prime?

  1. Oct 10, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove or disprove: Is 10^1,000 - 9 Prime?


    2. Relevant equations



    3. The attempt at a solution

    10^1,000 = 999...91.

    Is there a way to logically argue to drop the first nine hundred ninety eight 9's and just look at 91 as being a prime?
     
  2. jcsd
  3. Oct 10, 2013 #2

    Mark44

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    No.
    Notice that 101000 is a perfect square, and so is 9.
     
  4. Oct 10, 2013 #3

    SteamKing

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    You would think that might work.
    However:
    91 is prime
    991 is prime
    9991 = 103*97, but both of these are prime factors
    99991 is prime
    999991 = ?
     
  5. Oct 10, 2013 #4
    But the difference between two perfect squares isn't always prime. For example, 25-16=9.

    Not following the logic yet :/
     
  6. Oct 10, 2013 #5

    Mark44

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    What can you do with the difference of two squares?
     
  7. Oct 10, 2013 #6

    Office_Shredder

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    That's why he answered "No" to the question of whether it's prime!
     
  8. Oct 10, 2013 #7
    LOL he gave away the answer then!

    I've gotten to this point now: "Call 10^1,000 x^2 and 9=3^2. x2-32=(x+3)(x-3)."

    Thinking a proof by contradiction technique may work but mulling it over I can't see how (x+3)(x-3)=p, where p is prime, would lead to a contradiction. If I'm on the right path let me know and I'll try to work it out some more.
     
  9. Oct 10, 2013 #8

    Office_Shredder

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    You should just start writing down what you know about prime numbers, you should write down the relevant point fairly quickly.
     
  10. Oct 10, 2013 #9
    "...(x+3)(x-3)=N, and N is divisible by (x+3) OR (x-3). It cannot be prime since a prime is only divisible by itself and the number 1."

    that work?
     
  11. Oct 10, 2013 #10

    Office_Shredder

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    That sounds reasonable, except I'm not sure why you use the word OR when describing what N is divisible why.... and would be more appropriate.
     
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