Help with Solving Diophantine Equations: Is Brute-Force the Only Option?

[jobsism]

EDIT:- My question earlier was in error. I've edited the question accordingly and my required set of Diophantine equations is the one given below:-

For a problem that I'm working on, I need to solve the following system of Diophantine equations:-

$$a^3 + 40033 = d$$ $$b^3 + 39312 = d$$ $$c^3 + 4104 = d$$, where $a, b, c, d > 0$ are all DISTINCT positive integers and $a, b, c \notin$ {$2, 9, 15, 16, 33, 34$}.

How does one go about solving this? Is brute-force the only possible way? Or could there be a case that no integer solutions exist for this equation?

Also, are there any online computing engines, that allow me to set constraints, and solve Diophantine equations of this sort?

Any and all help is appreciated! Thanks!

 

[tiny-tim]

hi jobsism!

let's go for the easy equation …

a³ - b³ = 721 …

doesn't that make it obvious what a and b must be? ​

 

[Ryuzaki]

a³ - b³ = 721 …

doesn't that make it obvious what a and b must be? ​

Don't you mean $b^3 - a^3 = 721$, tiny-tim?

Your method is standard, but it requires checking and cross-checking for quite a number of values.

To the OP: In case you didn't get tiny-tim's hint, what he's trying to say is that $b^3 - a^3 = 721$ gives $(b-a)(a^2 + ab + b^2) = 7$x$103$, and so $(b-a) \in$ {$1,7,103,721$}. Then you compute $a$ and $b$ for each of these values, get the corresponding $d$ values, and plug it in the equation, $c^3 +4104 = d$. This is a rather tedious method in my opinion, but it would work.

However, I suggest a slightly easier approach. First, notice that for any solution, $b^3 - a^3≥(a+1)^3 - a^3 = 3a^2 + 3a + 1$. This gives a direct upper bound on $a$, namely $15$. Now, checking for which of $a = 1,2,3,...15$, one has that $a^3 + 721$ is the third power of an integer, one finds that this is only the case for $a=2$ and $a=15$ (where $b$ would be $9$ and $16$ respectively).

Both the above solutions are rejected, as per your constraints. So already, your first two equations do not have any solutions.

 

[tiny-tim]

Hi Ryuzaki!

To the OP: In case you didn't get tiny-tim's hint, what he's trying to say is that $b^3 - a^3 = 721$ gives $(b-a)(a^2 + ab + b^2) = 7$x$103$, and so $(b-a) \in$ {$1,7,103,721$}. Then you compute $a$ and $b$ for each of these values, get the corresponding $d$ values, and plug it in the equation, $c^3 +4104 = d$. This is a rather tedious method in my opinion, but it would work.

Slightly easier would be eg if b - a = 1, then (b - a)² - 3ab = 721, so 3ab = 720, ab = 240, (a,b) = (15,16), which the question doesn't allow

 

[CellCoree]

Hello,

Thank you for reaching out for help with solving your Diophantine equations. I understand the importance of finding efficient and accurate solutions to problems. In this case, brute-force may not be the only option for solving your system of Diophantine equations.

First, let's define what a Diophantine equation is. It is an equation where the solutions must be integers. In your case, the variables a, b, c, and d must be positive integers, and they must all be distinct. This means that they cannot have the same value.

Brute-force involves trying every possible combination of values for the variables until a solution is found. This can be time-consuming and may not always lead to a solution. However, there are other methods that can be used for solving Diophantine equations.

One method is to use algebraic techniques, such as factoring and substitution, to simplify the equations and find a solution. Another method is to use number theory, which involves studying the properties of integers and their relationships to find a solution.

It is also possible that no integer solutions exist for this particular system of Diophantine equations. This can be determined by analyzing the equations and their constraints.

As for online computing engines, there are several that can solve Diophantine equations, such as Wolfram Alpha and Mathway. These engines allow you to set constraints and input your equations to find solutions.

I hope this information helps you in solving your Diophantine equations. Best of luck to you in your research!