My Math Forum Diophantine equations.

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 November 15th, 2015, 08:36 PM #1 Member   Joined: Jul 2015 From: tbilisi Posts: 30 Thanks: 8 Diophantine equations. I don't quite understand the approach and discussion in this section. When I tried to discuss some Diophantine equations - all the messages in my deleted. I've been banned. At the same time quietly gets clearly crazy themes. So you have to use other resources. User individ - Mathematics Stack Exchange Formulas for the solution of Diophantine equations. (Page 1) / This is Cool / Math Is Fun Forum http://www.artofproblemsolving.com/community/c3046 Maybe there is some kind of specificity? What is considered good and what topics it will be in this section? explain to me. Why? Crazy explicit proof of Fermat's last theorem. Stir can. A solution of the equations?
 November 16th, 2015, 04:19 AM #2 Senior Member   Joined: Apr 2015 From: Barto PA Posts: 170 Thanks: 18 Following is a comment by one of your reviewers on the mathisfun site. "Then I don't understand why you post your solutions to begin with. Do you think that it's *impossible* for someone to figure out how you find these solutions? If not, then if your discovery means anything to you, you need to stop giving your results to people, because someone can come along and figure out your method and legally claim it as his own. Even if you are a genius, there are other geniuses out there. This is just algebra after all. Your choice to post solutions without explaining a method to get them is going to aggravate anyone who cares about the subject matter, as you are just *teasing* them. If doing this has become a highlight in your life, then by no means stop, but I am puzzled that you are in shock of the reactions you are getting. In fact, I'm surprised they didn't ban you yet for being a *spammer*, because that is exactly how you are behaving as far as forums are concerned." There's more to that comment but my not-so- smart phone would not copy it. Anyone who is interested can see page 9 of Individ's post: Formulas for the solution of Diophantine equations. (Page 9) / This is Cool / Math Is Fun Forum By the way: Did you really claim to have solved Hilbert's tenth? Congratulations. I think you will have a problem on this Forum too, but I'm gonna stay out of it.
 November 16th, 2015, 04:30 AM #3 Member   Joined: Jul 2015 From: tbilisi Posts: 30 Thanks: 8 Of course, all the equations cannot be solved. But many can be solved. Like for example the Legendre equation. Solving a Diophantine equation of the form $x^2 = ay^2 + byz + cz^2$ with the constants $a, b, c$ given and $x,y,z$ positive integers - Mathematics Stack Exchange Of course you can say that to solve anything. Declare 10 Hilbert's problem is unsolvable. But you can try to figure out what can be solved. And how to make it as easy as possible. My Blog is already 250 posts. Each post is at least one formula.
November 16th, 2015, 05:08 AM   #4
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 Originally Posted by Individ Of course, all the equations cannot be solved. But many can be solved. Like for example the Legendre equation. Solving a Diophantine equation of the form $x^2 = ay^2 + byz + cz^2$ with the constants $a, b, c$ given and $x,y,z$ positive integers - Mathematics Stack Exchange Of course you can say that to solve anything. Declare 10 Hilbert's problem is unsolvable. But you can try to figure out what can be solved. And how to make it as easy as possible. My Blog is already 250 posts. Each post is at least one formula.
So you are not dealing with Hilbert's 10th. You are just posting formulas. The problem is to prove whether there is a general algorithm to find rational roots in a finite number of steps.

This is not the same as finding algebraic formulas. An algorithm not necessarily is a closed-form formula. It could be e.g. an infinity of recursive formulas which have some predictable behavior but they would be such that their conjunction cannot be written in a single algebraic expression, although an algorithm could capture them.

Thus Matiyasevich's proof is profound since it shows there is no such procedure/algorithm, so as a corollary there is no such conjunction of "predictable" formulas applicable to any case given. Thus, I'm sorry to say, but your efforts are futile, if you'd live forever and would post infinite formulas to solve concrete Diophantine equations, it wouldn't still be anything decisive for Hilbert's 10th.

 November 16th, 2015, 05:24 AM #5 Member   Joined: Jul 2015 From: tbilisi Posts: 30 Thanks: 8 Well. If you think so, then we are not able to decide anything. One method and algorithm to solve infinitely many different equations. Not enough for You and still believe that it is impossible to solve. I understand that it is impossible to solve all. But there is much that can be done. And now. Because believe 10 the problem is not solvable. We must abandon altogether the attempt to solve the equations? If there is no General algorithm, as I found all these formulas? Each time coming up with a new algorithm?
November 16th, 2015, 05:36 AM   #6
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 Originally Posted by Individ And now. Because believe 10 the problem is not solvable. We must abandon altogether the attempt to solve the equations? (...) If there is no General algorithm, as I found all these formulas?
1. It is not "belief", is a formal proof;

2. I didn't say we should abandon anything;

3. You can find infinitely many formulas, of course, but you cannot find an unified procedure (think about it as a finite list of fixed procedures) so that it gives you ALL such formulas, so to speak.

Then, again, no "solution" to Hilbert's 10th Problem such efforts would represent.

On the other hand, yeah, of course that studying methods like that have some value. Maybe you find nice methods for a class of interesting equations no one knows.

 November 16th, 2015, 05:46 AM #7 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Individ, I don't know what you're claiming. Maybe you should start over by telling us what you're looking for, supposing that we haven't been following you on other forums.
November 16th, 2015, 06:29 AM   #8
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 Originally Posted by al-mahed 3. You can find infinitely many formulas, of course, but you cannot find an unified procedure (think about it as a finite list of fixed procedures) so that it gives you ALL such formulas, so to speak.
Formula permanently removed. On this forum were deleted 2 times.
So probably better to give a link to my Blog.
http://www.artofproblemsolving.com/community/c3046

Who cares - you can discuss a specific equation.
It's hard for me to understand. How you can solve as many equations having no General solution algorithm?

In this situation the best way is to mix solutions of certain equations. Where allows you to make some sort of forum.

As in the tale of 1000 and 1 night. Need more equations to solve, so appeared other opinion.

November 16th, 2015, 07:17 AM   #9
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 Originally Posted by Individ It's hard for me to understand. How you can solve as many equations having no General solution algorithm?
Many families of simple Diophantine equations have solutions. For example, here's one family for which an effective procedure is known: single equations with degree 2 or less with an arbitrary number of variables. (This includes, for example, the formula you posted at Nov 16, 2015, 2:18 am on your blog.) So unless some term is cubic or higher solutions can be found 'automatically'. For example, Dario's solver can handle almost all 2-variable Diophantine quadratics. Many other systems can solve these as well.

But more complicated Diophantine equations can be trickier, and in general the problem is unsolvable. The story of the proof is interesting, you should look it up.

Last edited by CRGreathouse; November 16th, 2015 at 07:19 AM.

 November 16th, 2015, 08:35 AM #10 Member   Joined: Jul 2015 From: tbilisi Posts: 30 Thanks: 8 That is what I'm saying. There are different approach. Do not use brute force, but to understand the phenomenon and the relationship. Need to solve the equation. The formulae can further be useful. The solutions of the equations can be unknowns in the other equations. And when just get the number - this will be a dead end for further calculations. Though it may be just entertainment. To write a useless formula.

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