System of quadratic Diophantine equations 4 3 x 2 + 4 3 x + 1...

Manteo2h

Manteo2h

Answered

2022-06-26

System of quadratic Diophantine equations
4 3 x 2 + 4 3 x + 1 = y 2
8 3 x 2 + 8 3 x + 1 = z 2

Answer & Explanation

Aaron Everett

Aaron Everett

Expert

2022-06-27Added 18 answers

We consider only the general question. It was proved by Matijasevich that there is no algorithm which, on input any Diophantine equation P ( x 1 , x 2 , , x m ) = 0, where P is a polynomial with integer coefficients, will determine whether the equation has an integer solution.
Using a little trick that goes back to Skolem, given any Diophantine equation P ( x 1 , x 2 , , x m ) = 0, we can algorithmically produce a system Q i ( y 1 , y 2 , , y n ) = 0, i = 1 , , s of quadratic Diophantine equations such that the system has a solution in integers if and only if P ( x 1 , x 2 , , x m ) = 0has a solution in integers.
It follows that there is no algorithm which, given any system of quadratic Diophantine equations, will determine whether the system has a solution in integers.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?