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January 9th, 2017, 06:05 PM   #1
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(ax^3+bx^2+cx+d) + (ay^3+by^2+cy+d) = (az^3+bz^2+cz+d)

Is there any way to find positive integer solutions to (ax^3+bx^2+cx+d) + (ay^3+by^2+cy+d) = (az^3+bz^2+cz+d), where a,b,c,d are some integers?
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January 9th, 2017, 07:34 PM   #2
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Note that for the special case $a = 1, b = c = d = 0$, the equation becomes $x^3 + y^3 = z^3$. Although that equation was proved (by Euler, I think) to have no positive integer solutions, the proof, while elementary, is not simple.

Is it possible that the same techniques could completely resolve your more general equation? Possibly, but I doubt it.

In any case, are you familiar with the proof that the equation $x^3 + y^3 = z^3$ has no positive integer solutions? Consider that as prerequisite knowledge for any meaningful investigation of your question.
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January 9th, 2017, 08:33 PM   #3
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Quote:
Originally Posted by Naveenchandra Kumar View Post
(ax^3+bx^2+cx+d) + (ay^3+by^2+cy+d) = (az^3+bz^2+cz+d)
Same as:
ax^3+bx^2+cx + ay^3+by^2+cy = az^3+bz^2+cz-d

Is that a TRUE problem. or did you make it up?
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January 9th, 2017, 11:05 PM   #4
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You want to get a simple answer. It can not be.
Parameterization itself looks very cumbersome - you do not want.

Here the deadlock - because you want what can not be.
Quite often hard to solve Diophantine equations. And besides, the solutions themselves look very bulky.

For this equation, the number of solutions for given factors - if any. The final number. Formula communication solutions with coefficients can not write - because it does not exist. Sami factors still have to parameterize.
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January 10th, 2017, 05:34 AM   #5
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Are you all suggesting that solution does not exist or method does not exist?

Are you all suggesting that solution does not exist or method does not exist?. Could you people be little more specific, I am novice in this field.
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January 10th, 2017, 05:45 AM   #6
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Yes sir, I made up this problem for curiosity.

Quote:
Originally Posted by Denis View Post
Same as:
ax^3+bx^2+cx + ay^3+by^2+cy = az^3+bz^2+cz-d

Is that a TRUE problem. or did you make it up?
Yes sir, I made up this problem for curiosity, after looking into $x^n+y^n=z^n$ and Fermat's Last Theorem.
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January 10th, 2017, 06:01 AM   #7
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Ok.
WHAT are the givens: a, b, c and d?

Do you realise that a and b are interchangeable:
if as example we get a solution with a=3 and b=5,
then a=5 and b=3 is also a solution.

No matter how you cut it, a cubic equation needs
to be solved: agree?

Here's a solution for x:
Wolfram|Alpha: Computational Knowledge Engine

Perhaps in the future you can go to that site to "see"
what your made-up equations mean...
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January 10th, 2017, 05:16 PM   #8
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some examples

Quote:
Originally Posted by Denis View Post
Ok.
WHAT are the givens: a, b, c and d?

Do you realise that a and b are interchangeable:
if as example we get a solution with a=3 and b=5,
then a=5 and b=3 is also a solution.

No matter how you cut it, a cubic equation needs
to be solved: agree?

Here's a solution for x:
Wolfram|Alpha: Computational Knowledge Engine

Perhaps in the future you can go to that site to "see"
what your made-up equations mean...
For example, given
$(x^3 + 3*x^2 + 2*x + 6) + (y^3 + 3*y^2 + 2*y + 6) = (z^3 + 3*z^2 + 2*z + 6)$, $a=1, b=3, c=2, d=6$, This has the following integer solutions, $(4,5,6), (5,7,8 ), (14,32,33), (25, 75, 76), (27, 84, 85), (8, 9, 11), (22, 43, 45), (24, 49, 51), (63, 207, 209), (17, 19, 23), (229, 707, 715), (117, 228, 238 )$.

Last edited by Naveenchandra Kumar; January 10th, 2017 at 05:54 PM.
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January 10th, 2017, 05:30 PM   #9
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Quote:
Originally Posted by Naveenchandra Kumar View Post
For example, given
$(x^3 + 3*x^2 + 2*x + 6) + (y^3 + 3*y^2 + 2*y + 6) = (z^3 + 3*z^2 + 2*z + 6), a=1, b=3, c=2, d=6$, This has the following integer solutions, (4,5,6), (5,7,, (14,32,33), (25, 75, 76), (27, 84, 85), (8, 9, 11), (22, 43, 45), (24, 49, 51), (63, 207, 209), (17, 19, 23), (229, 707, 715), (117, 228, 23.
OK. But what does this achieve?
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January 10th, 2017, 05:37 PM   #10
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Sir, are you serious?

Quote:
Originally Posted by Denis View Post
OK. But what does this achieve?
I mean, what can mathematics achieve? What is the achievement of Fermat's Last Theorem?.
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