Is 42 a sum of three cubes?

[Guest]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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"

[Guest]

Excellent Reference Textbook on Algebraic Geometry:

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

[Guest]

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]

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]

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.

'Sum of three cubes for 42 finally solved—using real life planetary computer',

https://phys.org/news/2019-09-sum-cubes-solvedusing-real-life.html.

Wow! [tex]x^{3}+y^{3}+z^{3}= 42[/tex] when x = -80,538,738,812,075,974; y = 80,435,758,145,817,515; and z = 12,602,123,297,335,631.

Relevant Reference Link:

https://www.wolframalpha.com/input/?i=%28+-80%2C538%2C738%2C812%2C075%2C974%29%5E3+%2B+%2880%2C435%2C758%2C145%2C817%2C515%29%5E3+%2B%2812%2C602%2C123%2C297%2C335%2C631%29%5E3&assumption=%22ListOrNumber%22+-%3E+%7B%22Number%22%2C+%2280%2C435%2C758%2C145%2C817%2C515%22%7D

[Guest]

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[Guest]

FYI:

[tex]569936821221962380720^{3} + (-569936821113563493509)^{3} + (-472715493453327032)^{3} = 3[/tex].

Source Link:

"After cracking the 'sum of cubes' puzzle for 42, researchers discover a new solution for 3."

https://phys.org/news/2021-03-sum-cubes-puzzle-solution.html.

[Guest]

A link verification for the sum of three squares equals three:

https://www.wolframalpha.com/input/?i=569936821221962380720%5E3+%2B+%28-569936821113563493509%29%5E3+%2B+%28-472715493453327032%29%5E3+.

[Guest]

A link verification for the sum of three cubes equals three:

https://www.wolframalpha.com/input/?i=569936821221962380720%5E3+%2B+%28-569936821113563493509%29%5E3+%2B+%28-472715493453327032%29%5E3+.