# Is 42 a sum of three cubes?

### Re: Is 42 a sum of three cubes?

Guest wrote:For the given problem, how do we adapt the Newton Method to find integral solutions (assuming their existence) as fast as possible?

We should also consider what we know and what we seek:

We have an integral solution to equation,

1. $$a^{3}+b^{3}+c^{3}= 33$$.

We seek an integral solution to equation,

E. $$x^{3}+y^{3}+z^{3}= 42$$.

Moreover, we have an integral solution to equation,

2. $$e^{3}+f^{3}+g^{3}= 52$$.

So between the solutions of equations, one and two, we should be able find a solution to equation E if it exists...

Dave.
Guest

### Re: Is 42 a sum of three cubes?

Guest wrote:
Guest wrote:For the given problem, how do we adapt the Newton Method to find integral solutions (assuming their existence) as fast as possible?

We should also consider what we know and what we seek:

We have an integral solution to equation,

1. $$a^{3}+b^{3}+c^{3}= 30$$.

We seek an integral solution to equation,

E. $$x^{3}+y^{3}+z^{3}= 42$$.

Moreover, we have an integral solution to equation,

2. $$e^{3}+f^{3}+g^{3}= 52$$.

So between the solutions of equations, one and two, we should be able find a solution to equation E if it exists...

Dave.

1. $$3, 982, 933, 876, 681^{3} − 636, 600, 549,515^{3}$$ $$- 3, 977, 505, 554, 546^{3}= 30$$.

2. $$60, 702, 901, 317^{3} + 23, 961, 292, 454^{3} −61, 922, 712, 865^{3}$$ = 52.
Guest

### Re: Is 42 a sum of three cubes?

Hmm. We can extrapolate or interpolate or by some creative method process the data of equations, one and two, to help us solve equation E.

However, our creative processing does not rule out the Newton Method for solving equation E. It may help us find a solution faster.
Guest

### Re: Is 42 a sum of three cubes?

Play with the Numbers and Have Fun...

Example:

$$(2n*b+1)^3-(2n*b)^3-b^3=k$$,

http://m.wolframalpha.com/input/?i=%282n*b%2B1%29%5E3-%282n*b%29%5E3-b%5E3%3Dk+over+Integers

We hope the above example inspires a breakthrough! Good luck!
Guest

### Re: Is 42 a sum of three cubes?

Guest wrote:What does the graph of $$x^{3}+y^{3}+z^{3}=1$$ look like?

Source:

https://www.quora.com/What-does-the-graph-of-x-3+y-3+z-3-1-look-like

What does the graph of $$x^{3}+y^{3}+z^{3}=42$$ look like?

See a plot at link below,

https://develop.open.wolframcloud.com/app/objects/28edc87e-7575-436d-a0b0-10ae36b39bcf#sidebar=compute
Guest

### Re: Is 42 a sum of three cubes?

Guest wrote:
Guest wrote:What does the graph of $$x^{3}+y^{3}+z^{3}=1$$ look like?

Source:

https://www.quora.com/What-does-the-graph-of-x-3+y-3+z-3-1-look-like

What does the graph of E. $$x^{3}+y^{3}+z^{3}=42$$ look like?

See a plot at link below,

https://develop.open.wolframcloud.com/app/objects/28edc87e-7575-436d-a0b0-10ae36b39bcf#sidebar=compute

The shape and smoothness of our graph equation E above strongly suggests there exists an integral solution for equation E.

Go Blue!

Dave
Guest

### Re: Is 42 a sum of three cubes?

Guest wrote:
Guest wrote:
Guest wrote:What does the graph of $$x^{3}+y^{3}+z^{3}=1$$ look like?

Source:

https://www.quora.com/What-does-the-graph-of-x-3+y-3+z-3-1-look-like

What does the graph of E. $$x^{3}+y^{3}+z^{3}=42$$ look like?

See a plot at link below,

https://develop.open.wolframcloud.com/app/objects/28edc87e-7575-436d-a0b0-10ae36b39bcf#sidebar=compute

The shape and smoothness of our graph of equation E above strongly suggests there exists an integral solution for equation E.

Go Blue!

Dave
Guest

### Re: Is 42 a sum of three cubes?

Guest wrote:Play with the Numbers and Have Fun...

Example:

$$(2n*b+1)^3-(2n*b)^3-b^3=k$$,

http://m.wolframalpha.com/input/?i=%282n*b%2B1%29%5E3-%282n*b%29%5E3-b%5E3%3Dk+over+Integers

We hope the above example inspires a breakthrough! Good luck!

Algebra and number theory, together, solves our problem. And they also accomplish much more...
Amen!
Guest

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