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Modulo a prime

  1. Sep 28, 2008 #1
    1. The problem statement, all variables and given/known data

    Show the equation x^3 + 2y^3 =5 has no solution for x,y in Z, by considering it modulo a prime

    2. Relevant equations

    3. The attempt at a solution

    I need help starting this problem, I've been stuck on it for a while and don't even have a clue of how to start it.
  2. jcsd
  3. Sep 29, 2008 #2


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    Staff Emeritus
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    When it says "modulo a prime" it does not mean you must prove it has no solutions modulo any prime. Just try a convenient prime. In particular, consider this modulo 5: it reduces to x3+ 2y[/sup]3[/sup]= 0. You could just do the calculations for all 25 pairs from (0,0) to (4,4) but I think you can just note that x3 and 2y3 are additive inverses. The additive inverse of 1 is 4 and the additive inverse of 2 is 3. Can one be x3 and the other 2y3? Of course, x= y= 0 does solve this equation (modulo 5) which tells you that any solution to the original equation must involve only multiples of 5.
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