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A couple of Number Theory questions

  1. May 9, 2012 #1
    1. Find all solutions x (with 0 ≤ x ≤ 96) to the congruence 13x^385 + 73x^304 + x^290 + 10x^193 + 24x^112 + 70x + 76 ≡ 0 (mod 97)

    I was able to reduce, using Fermat's Little Theorem, to get 97x^16 + x^2 + 93x + 76 ≡ 0 (mod 97), but I don't know how to proceed from there. Is there another trick I can use?


    2. http://imgur.com/a/DUyHC

    RR denotes two adjacent quadratic residues, while NN denotes two adjacent quadratic non-residues. RN is a residue followed by a non-residue, and NR is a non-residue followed by a residue. I tried to solve the problems by adding and subtracting the four expressions given in part b but I haven't made any progress.


    Thanks for the help!
     
    Last edited: May 9, 2012
  2. jcsd
  3. May 9, 2012 #2

    SammyS

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    One quick question: What is 97 mod 97 ?
     
  4. May 9, 2012 #3
    One quick response: UGHHHH sometimes I'm an idiot. Thanks!
     
  5. May 9, 2012 #4

    SammyS

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    Did you find the solution ?
     
  6. May 9, 2012 #5
    I sure did, I found two solutions just by using the quadratic formula on the equation that remains after the x^16 term goes to zero. Then I just had to make sure that they're between 0 and 97. My solutions aren't integers, which is sort of frustrating, but oh well.

    Any advice on the other problem?
     
  7. May 9, 2012 #6

    SammyS

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    Well, there are two integer solutions.

    What numbers are congruent to 93 and/or 76 (mod 97) ?
     
    Last edited: May 9, 2012
  8. May 9, 2012 #7
    Oh good call, thanks so much. I got integer solutions once I changed the x and the constant term to congruent values.
     
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