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Math Help - Find all integers m?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find all integers m?

    Find all integers m such that m + 3 and m^2 + 3m + 3 are perfect cubes?
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  2. #2
    Senior Member nikhil's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    Find all integers m such that m + 3 and m^2 + 3m + 3 are perfect cubes?
    let me give you an algorithm
    let m+3=a^3
    then m=a^3-3
    now put values of perfect cubes (its easy) in place of a^3.for each such value you will get an integral value of m.
    Do the same for m^2 + 3m + 3
    that is m^2 + 3m + 3-a^3=0
    solve this and insert different value of a^3.this time every solution may not be integral.
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  3. #3
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    Find all integers m such that m + 3 and m^2 + 3m + 3 are perfect cubes?
    i believe you want an m such that it holds simultaneously..

    i had a miscalculation a while ago but i found out the m=-2 is one..
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  4. #4
    Super Member fardeen_gen's Avatar
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    Yes, the question requires m which holds good for both. Which method did you use to arrive at m = -2?
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  5. #5
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    Did you find a solution to this? My 4 month old daughter is tired of waiting while I try to crack it!

    I have made a couple of observations.

    1. It is easy to show that m = -2 is a solution. This comes from the special case when m+3=a^3=b^3=m^2+3m+3. Solving for m gives m^2+2m=0 and consequently m= -2 or m = 0.

    2. If we let m+3=a^3 then the second equation can be written as m*a^3+3=b^3 which looks very much like the first equation.
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  6. #6
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    What are the integer solutions to y^2 = x^3 - 3 ?
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