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Unusual set of 3 integers

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  1. Feb 14, 2016 #1
    Has anyone else spotted an unusual set of three different integers A, B, & C such that
    A^n + B^n - C^n = A + B - C > 0 (n > 1 and A x B x C > 0)

    I leave the reader to see if they can find this set, or to ask me what they are.
     
  2. jcsd
  3. Feb 14, 2016 #2
    Im assuming here that the 'n' is real and not limited to an integer. Is this allowed or are there restrictions on 'n'?
     
  4. Feb 14, 2016 #3
    I should have said "an unusual set of four integers" as n is included. A search should find the set quite quickly since none of the integers is as high as 20.
     
  5. Feb 14, 2016 #4

    Samy_A

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    There are infinitely many solutions for n=2.
     
  6. Feb 14, 2016 #5
    For n = 2 surely there are no solutions. 3^2 + 4^2 - 5^2 = 0 3 + 4 - 5 = 2
     
  7. Feb 14, 2016 #6

    Samy_A

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    Maybe I misunderstand your question.
    Why is, for example, 4²+6²-7²=4+6-7=3 not a solution?
     
  8. Feb 14, 2016 #7

    Krylov

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    The product of ##A, B## and ##C## should be positive, I think.

    No, sorry, I misread. Your example should be fine.
     
  9. Feb 14, 2016 #8
    Sorry, you are correct. I should have said n > 2. I think that then there is only one set.
     
  10. Feb 14, 2016 #9

    Samy_A

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    No, there is more than one set.
     
  11. Feb 14, 2016 #10
    Many thanks for your speedy replies and research. My set is 16^5 + 13^5 - 17^5 = 12
    I'd be pleased to see what others you have discovered.
     
  12. Feb 14, 2016 #11

    Samy_A

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    ##35^3+119^3-120^3=35+119-120=34##
     
  13. Feb 14, 2016 #12
    Thanks for that.

    I actually found my set while searching for the least possible value for A^n + B^n - C^n which in the case of n = 3 the least possible = 2 (with A, B and C relatively prime) I get 64 for n = 4, 12 for n = 5, 69264 n=6, 2697354 n = 7
     
  14. Feb 14, 2016 #13

    PeroK

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    That makes it 3-0 to the Belgian.
     
  15. Feb 14, 2016 #14

    fresh_42

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    Or 15-3 in Rugby counts.
     
  16. Feb 15, 2016 #15
    Seems that these sets can only occur with power 1,2,3 and 5
     
  17. Feb 15, 2016 #16

    fresh_42

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    Boldings by me.
    You already ruled out 1 and 2.
    ???
     
  18. Feb 15, 2016 #17
    If I add the condition that A, B and C are to be relatively prime, then my set is probably unique
     
  19. Feb 15, 2016 #18

    Samy_A

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    Not quite, these also satisfy this additional condition:
    3,21,55,56
    3,31,56,59
    3,49,139,141
    3,85,91,111
    3,101,291,295
     
  20. Feb 15, 2016 #19

    fresh_42

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    8-0
    It's going to be a disaster ...
     
  21. Feb 15, 2016 #20
    It does not matter, he will keep adding conditions until his set is unique. :)
     
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