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Posers and Puzzles

Posers and Puzzles

  1. 03 Jun '04 01:52
    Can you find the two smallest whole numbers where the difference of their squares is a cube, and the difference of their cubes is a square?

    -Ray.
  2. 03 Jun '04 09:37 / 1 edit
    Guess you forgot the word 'different'?

    0^2-0^2=0^3
    0^3-0^3=0^2

    In case this is true, you could take 6 and 10:

    10^2-6^2 = 100-36 = 64 = 8^3
    10^3-6^3 = 1000-216 = 784 = 28^2

    Are these the smallest?
  3. Standard member royalchicken
    CHAOS GHOST!!!
    03 Jun '04 17:10
    1^3 - 0^3 = 1^2

    1^2 - 0^2 = 1^3
  4. 03 Jun '04 17:29
    Originally posted by royalchicken
    1^3 - 0^3 = 1^2

    1^2 - 0^2 = 1^3
    Yes; but that is quite trivial. Maybe we should add the requirement that there is no zero involved?
  5. 04 Jun '04 00:05
    The problem did request whole numbers. There seems to be ambiguous definitions for the set of whole numbers. In this case, the set of whole numbers was intended to be the set of positive integers.

    In this case, the correct numbers are indeed 6 and 10.

    -Ray.
  6. 04 Jun '04 07:48
    The set of whole numbers is: -inf,...,-3,-2,-1,0,1,2,3,...,inf
    Then there are the natural numbers: 0,1,2,3,...,inf
    You wanted the natural numbers^+: 1,2,3,...,inf

    Might be handy to remember; it stops lots of confusion
  7. Donation Acolyte
    Now With Added BA
    04 Jun '04 08:14
    Originally posted by piderman
    The set of whole numbers is: -inf,...,-3,-2,-1,0,1,2,3,...,inf
    Then there are the natural numbers: 0,1,2,3,...,inf
    You wanted the natural numbers^+: 1,2,3,...,inf

    Might be handy to remember; it stops lots of confusion
    I take the natural numbers or N to exclude zero - after all zero was not thought to be a number at all in the past, so it's hardly 'natural' from a human perspective, and it's not really a counting number either, unless you're a computer scientist. If I need to write 'the natural numbers with zero' I write Z subscript(at least 0) (looks better with the appropriate symbols). I suppose I could use Z subscript(strictly greater than 0) instead of N, as that would also be unambiguous.
  8. Standard member TheMaster37
    Kupikupopo!
    06 Jun '04 18:05
    Lol, piderman. You give the most trivial solution of all, assume the author meant different numbers, then make a point of someone giving the most trivial solution to your extra condition, you should be more precise in your conditions :p