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I Two interesting equalities

  1. Aug 9, 2017 #1
    A^p - B^p - C^p = A - B - C
    With p > 1 this appears to occur only when p = 5: A = 17: B = 16 : C = 13

    A^p - B^p - C^p = D^p - E^p - F^p = G^p - H^p - I^p = Y
    A,B,C = 3,2,1
    D,E,F = 9,8,7
    G,H,I = 37,36,21
    ( Y = 64)

    Are these really the only occurrences of these equalities?
     
  2. jcsd
  3. Aug 9, 2017 #2

    jbriggs444

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    Science Advisor

    You are asking about solutions where all variables take on positive integer values?
     
  4. Aug 9, 2017 #3

    mfb

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    2016 Award

    Staff: Mentor

    I'll assume A,B,C > 0, otherwise there are many trivial solutions.

    p=2 up to A=20:
    Code (Text):
    A   B   C
    7   6   4
    9   7   6
    11   10   5
    12   10   7
    14   11   9
    16   15   6
    17   14   10
    19   15   12
    There are many solutions for larger A as well.

    p=3 has many solutions as well, the smallest one is (16,15,9).

    p=4, p=5, p=6 and p=7 don't have a solution for A<200 apart from the one you posted. Heuristic arguments suggest solutions are very rare.
     
  5. Aug 9, 2017 #4
    I mean all variables to be positive integers greater than zero.
    Thanks for examples with p = 2 and 3, so let's change my question to have p > 3.
    The case A,B,C = 17,16,13 represents the nearest to Fermat being wrong with p = 5
     
  6. Aug 11, 2017 #5
    Also the three sets of A,B,C for p = 4 represent the least possible value of Y (A, B & C all different from each other)
     
  7. Aug 11, 2017 #6
    And with p = 1,2 or 3 the least possible value of Y is zero with an infinite number of sets (Pythagoras triples when p = 2, 1 or 2 with p = 3)
     
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