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

Question

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

Aaron Everett · Accepted Answer

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.

System of quadratic Diophantine equations 4 3 </mfrac> x 2 </msu

Answered question

Answer & Explanation

New Questions in Algebra I