System of quadratic Diophantine equations 4 3 x 2 + 4 3 x + 1...
Manteo2h
Answered
2022-06-26
System of quadratic Diophantine equations
Answer & Explanation
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 , where 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 , we can algorithmically produce a system , of quadratic Diophantine equations such that the system has a solution in integers if and only if has 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.