# On the Solution of Type One Diophantine Equations

### On the Solution of Type One Diophantine Equations

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

We are given the Type One Diophantine Equation,

1. $$AX^{M} = \sum_{i=1}^{l }A_{i }X_{i }^{M_{i }} = k$$,

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

### Re: On the Solution of Type One Diophantine Equations

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

We are given the Type One Diophantine Equation,

1. $$AX = \sum_{i=1}^{l }A_{i }X_{i }^{M_{i }} = k$$,

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. $$AX = \sum_{i=1}^{l }A_{i }X_{i }^{M} = k$$.

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

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

$$X_{i }^{M}$$ = {$$x_{i }^{m}, x_{i}^{m-1}, x_{i }^{m-2}, ..., x_{i}$$} $$\in \mathbb{Z}^{l}$$\$${0}^{l}$$ where $$1 \le i \le l$$.
Guest

### Re: On the Solution of Type One Diophantine Equations

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

### Re: On the Solution of Type One Diophantine Equations

\ge
Guest wrote: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. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} \ge 1$$,

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

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:\ge
Guest wrote: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 = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

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

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

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

3. $$N_{k+1} = N_{k} +J_{k}^{-1} * N_{k}$$ (Newton's Method).

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

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

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

### Re: On the Solution of Type One Diophantine Equations

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

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

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

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:
Guest wrote:\ge
Guest wrote: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 = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

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

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

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

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (Newton's Method).

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

viewtopic.php?f=63&t=7803

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

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

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

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

### Re: On the Solution of Type One Diophantine Equations

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

'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

### Re: On the Solution of Type One Diophantine Equations

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

'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

### Re: On the Solution of Type One Diophantine Equations

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

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

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

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

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (Newton's Method).

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

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

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote: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 = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

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

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

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

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (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.

'Graph Theory' by Prof. Keijo Ruohonen,

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

### Re: On the Solution of Type One Diophantine Equations

An Update:

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

Note: Some but not all integer coefficients, $$a_{inn }$$, associated with each variable, $$x_{i }$$, could be zero.
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:An Update:

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

Note: Some but not all integer coefficients, $$a_{inn }$$, associated with each variable, $$x_{i }$$, could be zero.
Guest

### Re: On the Solution of Type One Diophantine Equations

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

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

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

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

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

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (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.

'Graph Theory' by Prof. Keijo Ruohonen,

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

### Re: On the Solution of Type One Diophantine Equations

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

### Re: On the Solution of Type One Diophantine Equations

Guest wrote: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 = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

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

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

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

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

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (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.

'Graph Theory' by Prof. Keijo Ruohonen,

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

'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

### Re: On the Solution of Type One Diophantine Equations

Guest 