How to describe all integer solutions to?
by individ, Jun 26, 2025, 1:27 PM
[url]https://mathoverflow.net/questions/415173/how-to-describe-all-integer-solutions-to-x2y2-z31/496760#496760 [/url]
It is quite difficult to obtain parametrization of all solutions of the equation.

Therefore, I think it's worth making several attempts at a solution, it's quite possible that which one will give the result.
If we use solutions of the Pell equation.

Let's introduce the coefficients.


For solutions of the Pell equations of the form.




It is quite difficult to obtain parametrization of all solutions of the equation.

Therefore, I think it's worth making several attempts at a solution, it's quite possible that which one will give the result.
If we use solutions of the Pell equation.


Let's introduce the coefficients.


For solutions of the Pell equations of the form.




Integer solutions of....
by individ, Jun 25, 2025, 2:21 PM
https://mathoverflow.net/questions/203435/integer-solutions-of-x2-48y213z2/496717#496717
Let's look at the general case.

When the coefficients of this equation are not represented in the form of squares. Then a funny pattern emerges.
If we know some solution to the equation and we know any solution to the Pell equation. Then we can always construct the following solution to this equation.

Formally, the relationship between the solvability of the Lagrange equation in general and the solvability of the equivalent Pell equation is obtained.
Numbers can have any signs.



Solutions to the general equation.

They are found using solutions.


Consider the example mentioned in the question.



Solutions to the Pell equation


We substitute it and get it. The first gives
, the second 
For any value of
, an infinite number of solutions are obtained.
Let's look at the general case.

When the coefficients of this equation are not represented in the form of squares. Then a funny pattern emerges.
If we know some solution to the equation and we know any solution to the Pell equation. Then we can always construct the following solution to this equation.

Formally, the relationship between the solvability of the Lagrange equation in general and the solvability of the equivalent Pell equation is obtained.
Numbers can have any signs.



Solutions to the general equation.

They are found using solutions.


Consider the example mentioned in the question.



Solutions to the Pell equation



We substitute it and get it. The first gives


For any value of

A variant of the solution of the general quadratic equation.
by individ, Jun 25, 2025, 10:22 AM
https://math.stackexchange.com/questions/4624896/all-natural-number-solutions-for-the-equation-a2b2-2c2
Let's look at the general case.

When the coefficients of this equation are not represented in the form of squares. Then a funny pattern emerges.
If we know some solution to the equation and we know any solution to the Pell equation. Then we can always construct the following solution to this equation.



Formally, the relationship between the solvability of the Lagrange equation in general and the solvability of the equivalent Pell equation is obtained. Numbers can have any signs.
Let's look at the general case.

When the coefficients of this equation are not represented in the form of squares. Then a funny pattern emerges.
If we know some solution to the equation and we know any solution to the Pell equation. Then we can always construct the following solution to this equation.



Formally, the relationship between the solvability of the Lagrange equation in general and the solvability of the equivalent Pell equation is obtained. Numbers can have any signs.
This post has been edited 1 time. Last edited by individ, Jun 25, 2025, 1:14 PM
Polynomial parametrization for solutions of quadratic Diophantine equations
by individ, Jun 19, 2025, 9:16 AM
https://mathoverflow.net/questions/425040/polynomial-parametrization-for-solutions-of-quadratic-diophantine-equations
On the solvability of the equation.

Any parameterization , if we want it to be complete. It must describe all the numbers, solutions for example for
. It must describe all the numbers from 
At the same time,
and
must be independent of each other. And the only parameter that depends on the rest is
.
Factorize the number.




On the solvability of the equation.

Any parameterization , if we want it to be complete. It must describe all the numbers, solutions for example for


At the same time,



Factorize the number.





The Markov equation in general form
by individ, Jun 18, 2025, 4:49 PM
https://mathoverflow.net/questions/392993/on-markoff-type-diophantine-equation
It is worth mentioning the method of finding solutions to a general equation.

Since this is the general Pell equation. Then all solutions can be found by knowing the solutions of the following equations.


The solutions can be written in this form.


To build solutions, you need to know the solutions.

Knowing any two solutions.


The rest of the solutions can be obtained.


It is worth mentioning the method of finding solutions to a general equation.

Since this is the general Pell equation. Then all solutions can be found by knowing the solutions of the following equations.


The solutions can be written in this form.


To build solutions, you need to know the solutions.

Knowing any two solutions.


The rest of the solutions can be obtained.


This post has been edited 2 times. Last edited by individ, Jun 18, 2025, 4:53 PM
Congruent numbers
by individ, Jun 17, 2025, 5:32 AM
https://math.stackexchange.com/questions/1576861/on-p2-nq2-z2-p2-nq2-t2-and-the-congruent-number-problem
If we know of any solution to the system.

Then for the system

You can write a solution




If we know of any solution to the system.

Then for the system

You can write a solution




Erdős-Straus conjecture
by individ, Jun 15, 2025, 9:36 AM
[url]https://math.stackexchange.com/questions/450280/erdős-straus-conjecture/5075634#5075634[/url]
Still, it's worth considering the general case.

I gave a lot of formulas, but all of them somehow boiled down to the need to factor
. And the solution somehow boils down to this form.

The task actually boils down to finding a single number.

And getting solutions is going through all the possible options. Going through all the options does not require a lot of calculations. I'll show you an example of the question I was asked.
The meaning of the search is in another record. Separate the whole part and consider the case when the remainder is whole.

I'll give you an example.
Let.


Let.

In this case, it is enough to check the numbers.
There is no point in checking further. Because the remainder becomes less than 1, and we only need integers.
And the solution will be numbers.


Still, it's worth considering the general case.

I gave a lot of formulas, but all of them somehow boiled down to the need to factor


The task actually boils down to finding a single number.

And getting solutions is going through all the possible options. Going through all the options does not require a lot of calculations. I'll show you an example of the question I was asked.
The meaning of the search is in another record. Separate the whole part and consider the case when the remainder is whole.

I'll give you an example.

Let.




Let.


In this case, it is enough to check the numbers.

There is no point in checking further. Because the remainder becomes less than 1, and we only need integers.
And the solution will be numbers.






The sum of any two elements is a square
by individ, Jun 12, 2025, 5:09 PM
https://math.stackexchange.com/questions/952801/biggest-set-such-that-sum-of-any-pair-is-perfect-square/952806#952806
After the publication of the solution in this form, there were many questions about the too cumbersome formula. It turns out that many people may not like the formula.
Therefore, I made an attempt to find a much easier parameterization of the solutions of this system.

The numbers can be found using the formulas




The parameterization of the solution can be written as.







I understand that there are two questions. This is the desire of some to use a computer to sort through all the solutions and present the solutions in a different form. Therefore, it is worth adding a few words.
Expressing numbers in terms of squares, we formulate the problem in a different way. We are looking for the number
, which can be represented as the sum of squares in 3 different ways.

The formula is known and the solution can be written down.
https://artofproblemsolving.com/community/c3046h1359115_representation_of_a_number_the_squares_in_different_ways_2






After the publication of the solution in this form, there were many questions about the too cumbersome formula. It turns out that many people may not like the formula.
Therefore, I made an attempt to find a much easier parameterization of the solutions of this system.

The numbers can be found using the formulas




The parameterization of the solution can be written as.







I understand that there are two questions. This is the desire of some to use a computer to sort through all the solutions and present the solutions in a different form. Therefore, it is worth adding a few words.
Expressing numbers in terms of squares, we formulate the problem in a different way. We are looking for the number


The formula is known and the solution can be written down.
https://artofproblemsolving.com/community/c3046h1359115_representation_of_a_number_the_squares_in_different_ways_2






This post has been edited 2 times. Last edited by individ, Jun 16, 2025, 4:11 PM
A system of similar nonlinear equations
by individ, Dec 29, 2022, 12:45 PM
There is a whole group of such systems of equations.

Such a system is solved as standard. First, we write down the parametrization of one equation.





And then we find the parameterization for the necessary parameters.





Such a system is solved as standard. First, we write down the parametrization of one equation.





And then we find the parameterization for the necessary parameters.




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