Is 42 a sum of three cubes?

Re: Is 42 a sum of three cubes?

Postby Guest » Sun Apr 14, 2019 6:50 pm

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. [tex]a^{3}+b^{3}+c^{3}= 33[/tex].

We seek an integral solution to equation,

E. [tex]x^{3}+y^{3}+z^{3}= 42[/tex].


Moreover, we have an integral solution to equation,

2. [tex]e^{3}+f^{3}+g^{3}= 52[/tex].

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?

Postby Guest » Sun Apr 14, 2019 7:50 pm

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. [tex]a^{3}+b^{3}+c^{3}= 30[/tex].

We seek an integral solution to equation,

E. [tex]x^{3}+y^{3}+z^{3}= 42[/tex].


Moreover, we have an integral solution to equation,

2. [tex]e^{3}+f^{3}+g^{3}= 52[/tex].

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



Dave.


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


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

Re: Is 42 a sum of three cubes?

Postby Guest » Sun Apr 14, 2019 8:10 pm

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?

Postby Guest » Mon Apr 15, 2019 6:46 pm

Play with the Numbers and Have Fun...

Example:

[tex](2n*b+1)^3-(2n*b)^3-b^3=k[/tex],

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?

Postby Guest » Tue Apr 16, 2019 7:17 pm

Guest wrote:What does the graph of [tex]x^{3}+y^{3}+z^{3}=1[/tex] 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 [tex]x^{3}+y^{3}+z^{3}=42[/tex] 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?

Postby Guest » Tue Apr 16, 2019 7:31 pm

Guest wrote:
Guest wrote:What does the graph of [tex]x^{3}+y^{3}+z^{3}=1[/tex] 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. [tex]x^{3}+y^{3}+z^{3}=42[/tex] 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?

Postby Guest » Tue Apr 16, 2019 7:31 pm

Guest wrote:
Guest wrote:
Guest wrote:What does the graph of [tex]x^{3}+y^{3}+z^{3}=1[/tex] 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. [tex]x^{3}+y^{3}+z^{3}=42[/tex] 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?

Postby Guest » Wed Apr 17, 2019 1:18 am

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

Example:

[tex](2n*b+1)^3-(2n*b)^3-b^3=k[/tex],

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?

Postby Guest » Wed Apr 24, 2019 7:05 pm

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

Example:

[tex](2n*b+1)^3-(2n*b)^3-b^3=k[/tex],

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?

Postby Guest » Sat May 04, 2019 2:57 pm

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

Example:

[tex](2n*b+1)^3-(2n*b)^3-b^3=k[/tex],

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
(381.13 KiB) Downloaded 31 times
Guest
 

Re: Is 42 a sum of three cubes?

Postby Guest » Sun May 05, 2019 2:44 pm

Excellent Reference Textbook on Algebraic Geometry:

'Algebraic Geometry' by Prof. J.S. Milne (see attached file for details).
Attachments
Algebraic Geometry.pdf
(2 MiB) Downloaded 27 times
Guest
 

Re: Is 42 a sum of three cubes?

Postby Guest » Tue May 07, 2019 12:48 pm

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?

Postby Guest » Tue May 07, 2019 1:49 pm

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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