# 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

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

Guest wrote:
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!

The algebraic curves have solutions galore. Seek them and you shall have glory! Amen!
Guest

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

Guest wrote:
Guest wrote:
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!

The algebraic curves have solutions galore. Seek them and you shall have glory! Amen!

Study the works of Emmy Noether (https://en.wikipedia.org/wiki/Emmy_Noether), the great master of Algebra:

"On Emmy Noether and Her Algebraic Works"
Attachments
On Emmy Noether and Her Algebraic Works.pdf
Guest

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

Excellent Reference Textbook on Algebraic Geometry:

'Algebraic Geometry' by Prof. J.S. Milne (see attached file for details).
Attachments
Algebraic Geometry.pdf
Guest

### 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?

And of course, I believe our problem is solved by now.

However, the serious and obvious problem with the Newton Method with regards to our problem is that it does not distinguish between integer solutions and non-integer solutions... And there are infinitely many non-integer solutions to our problem. This approach... is unacceptable. We need a remedy (effective method).
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?

And of course, I believe our problem is solved by now.

However, the serious and obvious problem with the Newton Method with regards to our problem is that it does not distinguish between integer solutions and non-integer solutions... And there are infinitely many non-integer solutions to our problem. This approach... is unacceptable. We need a remedy (effective method).

I have an idea which may help us to solve our problem or similar or harder problems. I will discuss it soon under the topic, 'On the Solution of Type One Diophantine Equations'.

Dave.
Guest

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