On the Solution of Type One Diophantine Equations

[Guest]

'Entia non multiplicanda praeter necessitatem (Entities should not be multiplied beyond necessity)." -- Occam.

We are given the Type One Diophantine Equation,

1. [tex]AX^{M} = \sum_{i=1}^{l }A_{i }X_{i }^{M_{i }} = k[/tex],

and we ask the following questions:

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Question: What is a solution of equation one?

Question: How many solutions does equation one have?
...

Dave
https://www.researchgate.net/profile/David_Cole29

[Guest]

'Entia non multiplicanda praeter necessitatem (Entities should not be multiplied beyond necessity)." -- Occam.

We are given the Type One Diophantine Equation,

1. [tex]AX = \sum_{i=1}^{l }A_{i }X_{i }^{M_{i }} = k[/tex],

and we ask the following questions:

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Question: What is a solution of equation one?

Question: How many solutions does equation one have?
...

Dave
https://www.researchgate.net/profile/David_Cole29

Notes:

Update: 1. [tex]AX = \sum_{i=1}^{l }A_{i }X_{i }^{M} = k[/tex].

A is a diagonal matrix such A = {[tex]a_{ij }= 0[/tex] if [tex]i \ne j[/tex], otherwise [tex]a_{ij }[/tex] is a nonzero integer}.

X is a column vector such that X = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\[tex]{0}^{l}[/tex].

[tex]X_{i }^{M}[/tex] = {[tex]x_{i }^{m}, x_{i}^{m-1}, x_{i }^{m-2}, ..., x_{i}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\[tex]{0}^{l}[/tex] where [tex]1 \le i \le l[/tex].

[Guest]

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

[Guest]

\ge

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We formulate the following equation,

2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} \ge 1[/tex],

over the set on nonzero integers such that equation one has at least one nonzero integer solution.
...

Dave.

[Guest]

\ge

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = k[/tex] (Updated).

2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex].

over the set on nonzero integers such that equation one has at least one nonzero integer solution.

3. [tex]N_{k+1} = N_{k} +J_{k}^{-1} * N_{k}[/tex] (Newton's Method).


Dave.

[Guest]

We also think some knowledge/tools on algebraic curves will help us solve our problem...

Relevant Reference Link:

https://en.wikipedia.org/wiki/Algebraic_curve

[Guest]

Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

viewtopic.php?f=63&t=1793&start=60

[Guest]

\ge

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = k[/tex] (Updated).

2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex].

over the set on nonzero integers such that equation one has at least one nonzero integer solution.

3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (Newton's Method).

Relevant Reference Links:

https://www.math10.com/forum/viewtopic.php?f=63&t=8285

viewtopic.php?f=63&t=7803

Dave.

[Guest]

Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

https://www.math10.com/forum/viewtopic.php?f=63&t=1793

[Guest]

Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

https://www.math10.com/forum/viewtopic.php?f=63&t=1793

FYI: 'Solving Diophantine Equations' by Profs. Octavian Cira and Florentin Smarandache:

http://fs.unm.edu/SolvingDiophantineEquations.pdf

[Guest]

Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

https://www.math10.com/forum/viewtopic.php?f=63&t=1793

FYI: 'Solving Diophantine Equations' by Profs. Octavian Cira and Florentin Smarandache:

http://fs.unm.edu/SolvingDiophantineEquations.pdf

FYI:

'An Introduction to Diophantine Equations: A Problem-Based Approach' by Prof. T. Andreescu et al.

[Guest]

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...[/quote][/quote]

We think the solutions, X = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = \sum_{i=1}^{j_{1 }}A_{1}X_{1 }^{M} + \sum_{i=1}^{j_{2 } }A_{2}X_{2 }^{M} + ...[/tex]
[tex]= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1j_{1}j_{1}}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2j_{2}j_{2}}x_{2} + ... = k[/tex] Updated);

2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex]

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (Newton's Method).

Relevant Reference Links:

https://www.math10.com/forum/viewtopic.php?f=63&t=8285

https://www.math10.com/forum/viewtopic.php?f=63&t=7803

Dave.

[Guest]

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = \sum_{i=1}^{j_{1 }}A_{1}X_{1 }^{M} + \sum_{i=1}^{j_{2 } }A_{2}X_{2 }^{M} + ...[/tex]
[tex]= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1j_{1}j_{1}}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2j_{2}j_{2}}x_{2} + ... = k[/tex] Updated);

2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex]

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (Newton's Method).

...

Dave.

Hmm. Because of equation 2 and because our need for intelligent (efficient and directed) searches for solutions to equation one, we recommend some study of graph theory too.

Relevant Reference Link:

'Graph Theory' by Prof. Keijo Ruohonen,

http://math.tut.fi/~ruohonen/GT_English.pdf

[Guest]

An Update:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||\sum_{i=1}^{j_{1 }}A_{1}X_{1 }^{M}|| + ||\sum_{i=1}^{j_{2 } }A_{2}X_{2 }^{M}|| + ...[/tex]
[tex]= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1j_{1}j_{1}}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2j_{2}j_{2}}x_{2} + ... = k[/tex].

Note: Some but not all integer coefficients, [tex]a_{inn }[/tex], associated with each variable, [tex]x_{i }[/tex], could be zero.

[Guest]

An Update:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||A_{1}X_{1 }^{M}|| + ||A_{2}X_{2 }^{M}|| + ... + ||A_{l}X_{l }^{M}||[/tex]
[tex]= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1mm}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2mm}x_{2} + ... + a_{l11 }x_{l }^{m} + a_{l22 }x_{l }^{m-1} + ... + a_{lmm}x_{l} = k[/tex].

Note: Some but not all integer coefficients, [tex]a_{inn }[/tex], associated with each variable, [tex]x_{i }[/tex], could be zero.

[Guest]

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||A_{1}X_{1 }^{M}|| + ||A_{2}X_{2 }^{M}|| + ... + ||A_{l}X_{l }^{M}||[/tex]
[tex]= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1mm}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2mm}x_{2} + ... + a_{l11 }x_{l }^{m} + a_{l22 }x_{l }^{m-1} + ... + a_{lmm}x_{l} = k[/tex].

Note: Some but not all integer coefficients, [tex]a_{inn }[/tex], associated with each variable, [tex]x_{i }[/tex], could be zero.

2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex]

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (Newton's Method).

...

Hmm. Because of equation 2 and because our need for intelligent (efficient and directed) searches for solutions to equation one, we recommend some study of graph theory too.

Relevant Reference Link:

'Graph Theory' by Prof. Keijo Ruohonen,

http://math.tut.fi/~ruohonen/GT_English.pdf

[Guest]

Since my work in mathematics has brought me no happiness nor awards, I have decided to retire from it today…

Goodbye and good luck to all,

David Cole,

https://www.researchgate.net/profile/David_Cole29

[Guest]

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||A_{1}X_{1 }^{M}|| + ||A_{2}X_{2 }^{M}|| + ... + ||A_{l}X_{l }^{M}||[/tex]
[tex]= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1mm}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2mm}x_{2} + ... + a_{l11 }x_{l }^{m} + a_{l22 }x_{l }^{m-1} + ... + a_{lmm}x_{l} = k[/tex].

Note: Some but not all integer coefficients, [tex]a_{inn }[/tex], associated with each variable, [tex]x_{i }[/tex], could be zero.

2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex]

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (Newton's Method).

...

Hmm. Because of equation 2 and because of our need for intelligent (efficient and directed) searches for solutions to equation one, we recommend some study of graph theory too.

Relevant Reference Link:

'Graph Theory' by Prof. Keijo Ruohonen,

http://math.tut.fi/~ruohonen/GT_English.pdf

Relevant Reference Link:

'THE INSOLUBILITY OF CLASSES OF DIOPHANTINE EQUATIONS' By Profs., N. C. Ankeny and P. Erdös,

https://pdfs.semanticscholar.org/cee7/94fa9154cd0b4ac9d3c453f85918bf732bb1.pdf.

Remark: This paper is quite insightful! Enjoy!

[Guest]

FYI: 'ON THE INSOLUBILITY OF A CLASS OF DIOPHANTINE EQUATIONS AND THE NONTRIVIALITY OF THE CLASS
NUMBERS OF RELATED REAL QUADRATIC FIELDS OF RICHAUD-DEGERT TYPE' by Prof. R. A. MOLLIN.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A261C2B85FC9472F14D217BD71DE5B65/S0027763000000738a.pdf/div-class-title-on-the-insolubility-of-a-class-of-diophantine-equations-and-the-nontriviality-of-the-class-numbers-of-related-real-quadratic-fields-of-richaud-degert-type-div.pdf.