Finding Solutions to a System of Diophantine Equations

Mathematics Asked by GodOrGovern on November 26, 2020

I’m trying to find triplets of integer $$(x, y)$$ pairs – $$(x_1, y_1), (x_2, y_2), (x_3, y_3)$$ – that satisfy the following equations:

$${x_1}^2 + {y_1}^2 = {x_2}^2 + {y_2}^2 = {x_3}^2 + {y_3}^2 \ x_1 + x_2 + x_3 = 5 \ y_1 + y_2 + y_3 = 0 \ (x_1, y_1) neq (x_2, y_2) \ (x_1, y_1) neq (x_3, y_3) \ (x_2, y_2) neq (x_3, y_3)$$
Currently, for each integer $$c$$ that can be written as the sum of two squares, I create a list of all possible integer pairs $$(a, b)$$ for which $$a^2 + b^2 = c$$ and then check all possible triplets from each list for validity (actually I only need to check all possible combinations of 2 pairs, but the idea is the same). The only constraints I’ve found are $$c$$ must be divisible by 5 and $$a$$ and $$b$$ must have different parity. This is pretty inefficient, as only a very small subset of possible values of $$c$$ produce valid triplets. Checking the validity of any given triplet is relatively easy, so I’m hoping someone can help me find further constraints on the values of $$c$$. It is also entirely possible that there is some other, more efficient method of finding these triplets. Either way, help would be much appreciated.

Related Questions

How to prove that if $int_L vec{F}(vec{x})cdot dvec{x}$ exists, then integral on separate curve exists?

1  Asked on November 21, 2021 by shore

How to prove that these functions do not intersect?

1  Asked on November 21, 2021 by user805770

Evaluate: $int_0^1 sqrt{x+sqrt{x^2+sqrt{x^3+cdots}}}, dx.$

1  Asked on November 21, 2021

Find the dimension of $V = {f in C^k [0, 1] : a_n f^{(n)}(t)+cdots+a_0 f (t) = 0 textrm{ for all } t in [0, 1]}$.

0  Asked on November 21, 2021 by rajesh-sri

Periodic functions for the definite integral

1  Asked on November 21, 2021

Is $(a/b)-1$ approximately equal to $log_e (a/b)$

3  Asked on November 21, 2021 by noi-m

Solving $dx/dt=0.2x^2left(1-x/3right)$ for the initial condition $x(0)=x_0$

1  Asked on November 21, 2021

On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{frac{k-1}{2}} n^2$

1  Asked on November 21, 2021 by jose-arnaldo-bebita-dris

Help finding a centre of a circle

5  Asked on November 21, 2021 by billybob3234

Does local cohomology commutes with direct sums?

2  Asked on November 21, 2021 by ivon

Section 2 E, problem (b) Kelley

1  Asked on November 21, 2021

Sum involving the set of all possible combinations with at most two repetitions

2  Asked on November 21, 2021 by sharik

Showing a basis for polynomials

1  Asked on November 20, 2021 by duncank3

Examples of closed manifolds?

3  Asked on November 20, 2021

How to solve this ODE: $x^3dx+(y+2)^2dy=0$?

4  Asked on November 20, 2021

Non-negative convergent series $a_n$ where $limsup na_n >0$

1  Asked on November 20, 2021 by akm

Find $int_0^{frac{pi}{2}} e^{-a(sin(x)+cos(x))} , dx$ and/or $int_0^{frac{pi}{2}} e^{-a(sin(x)+cos(x))} sin(cos(x)) , dx$

0  Asked on November 20, 2021

Which integer combinations of $n$-th roots of unity are zero?

1  Asked on November 20, 2021

A problem of matrix on the equation $x_1+x_2+cdots+x_n=d$

0  Asked on November 20, 2021 by lsr314

How to compare Dehn Invariants

0  Asked on November 20, 2021 by mandelbroccoli