# a^3 + b^3 = c^3

by Loren Booda
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 P: 3,394 Does the equation a3 + b3 = c3 (where a, b and c are constants) have any general geometrical significance?
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 Quote by Loren Booda Does the equation a3 + b3 = c3 (where a, b and c are constants) have any general geometrical significance?
Not sure. What about if we give a triangle depth and do something like the Pythagoras theorem but use area instead of length?
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 Quote by John Creighto What about if we give a triangle depth and do something like the Pythagoras theorem but use area instead of length?
No way!

I don't think cubes ever arise in geometry.

Maybe in calculus, and in some branches of physics, but not in geometry.

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## a^3 + b^3 = c^3

Are you interested about values $a,b,c\in\mathbb{N}$ or $\in\mathbb{R}$?

With natural numbers, there is an equivalent problem with small cubes of fixed size. Suppose you have c^3 small cubes, piled into a one bigger cube. Can you take these c^3 small cubes, and form two new big cubes that contain precisely all these small cubes? The Fermat's last theorem says that you cannot.

With real numbers, an equation x^3+y^3=1 describes a surface of a L^p ball with value p=3. Frankly, I still have not understood why real analysis deals so much with L^p spaces. I don't know what's their significance, yet
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 Quote by tiny-tim No way! I don't think cubes ever arise in geometry. Maybe in calculus, and in some branches of physics, but not in geometry.
Well, so much for solid geometry!
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 Quote by tiny-tim No way! I don't think cubes ever arise in geometry.
 PF Patron HW Helper Sci Advisor Thanks P: 25,521 oh, those cubes! 
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 Quote by tiny-tim oh, those cubes! 
Or spheres.
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 P: 13 Ok tell me if theres a problem with this but I think this is the only way to solve it. cube root 5^3+cube root 5^3= cube root 10^3 the cube root just cancels out the cubing so the a and b variables need to be the same and the c variable needs to be double the a or b. I don't see what use this would have but thats the only way I've found to solve it.
 P: 13 I'm not bananaman, but a friend and I think his reasoning is flawed somewhere but the equation seems reasonable. I still believe that the equation is universally unsolvable, basically forbidden. WC
 PF Patron HW Helper Sci Advisor Thanks P: 25,521 How many Bananamen are there? Where is Talleyman when we need him!
 P: 70 There is no solution. No 2 (natural number) cubes add up to another cube. In fact no solutions exist in x^n + y^n =z^n, where n is any number greater then 2, as stated by Fermat's last theorem (as proved by Andrew Wiles). bananana you can't cancel out like this. One way to show this is wrong is 5^3 +5^3= 10^3 125+125=1000 250=1000 and this is clearly not true.
 P: 13 I'm still not seeing why a cube root number wouldn't cancel out the cube. I know it has no practical application but that equation i gave would work. what you left out in your proof is the cube root so it should be the (cube root of 5) to the third power. This also got me thinking about the ratio of three dementional object sides to each other, does anyone know some equations along with their application in regards to shape.
 P: 13 o and the "other" bananaman was just my friend using my account
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 Quote by Bananaman3579 I'm still not seeing why a cube root number wouldn't cancel out the cube. I know it has no practical application but that equation i gave would work. what you left out in your proof is the cube root so it should be the (cube root of 5) to the third power. This also got me thinking about the ratio of three dementional object sides to each other, does anyone know some equations along with their application in regards to shape.
Ok sorry I didn't see that bit, but regardless it's the same.

lets say you can cancel out the cube then

a=x^1\2
b=y^1/2
c=z^1/2

then a^3 +b^3= c^3

according to you a+b=c

so lets say 1+1=2

then plug back into the formula you get 1+1=8 which is not true

the cube rrot is irrelevantly
 P: 13 ok so I actually did this on a calculator, feel free to do so yourself, and the equation works. Now the problem i see with it is that it takes away from the "spirit" of the equation in the sense that it is just canceling out the main part of the equation and therefore basically destroying it. It reminds me of those verbal equations kids would say like, give me your favorite number then multiply by 2 add this and that, and the number would end up being the original through inverse operations. Basically i proved that there is a way to solve it but its pointless.
 P: 13 ok you might also have missed the fact that the number that had a cube root was the variable, a= cube root of one b= cube root of 1 c= cube root of 2. then plug that into the equation.