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    xtiancaz's Avatar
    xtiancaz Posts: 1, Reputation: 1
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    #1

    Feb 24, 2017, 05:38 AM
    If (a,b,c) are pythagorean triples such as a^2+b^2=c^2 then prove a^3+b^3+3ab>c^3
    If (a,b,c) are pythagorean triples such as a^2+b^2=c^2 then prove a^3+b^3+3ab>c^3

    Blocked on this for the past hour

    I understand they must be positive numbers

    Clearly c>a and c>b

    but and thinking answer around with using (a+b)^3=a3 + 3a2b + 3ab2 + b3

    or
    (a+b+c)^3 = a^3+b^3+c^3 + 3(a+b)(b+c)(a +c)

    but can't find the trick. Anyone can help? Thanks!


    ebaines's Avatar
    ebaines Posts: 12,129, Reputation: 1307
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    #2

    Feb 25, 2017, 07:28 AM
    Check your formula - it is incorrect. It works for the Pythagorean triplet 3,4,5:



    but not for the triplet 5,12,13:



    EDIT - I wonder if the correct formulation of this problem should be to prove that:

    ?? That would be a breeze (and applies to ALL triangles, not just right angle triangles with integer sides).

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