Why do we need the minimum solution of (A, B) with respect to some funtion in Vieta jumping?The minimal solution (A, B) with respect to some function of A and B, usually A + B, is taken. The equation is then rearranged into a quadratic with coefficients in terms of B, one of whose roots is A, and Vieta's formulas are used to determine the other root to the quadratic.However, I do not fully see the need to take the minimal solution, can someone explain that step for me?

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Why do we need the minimum solution of (A, B) with respect to some funtion in Vieta jumping?The minimal solution (A, B) with respect to some function of A and B, usually   A  +  B, is taken. The equation is then rearranged into a quadratic with coefficients in terms of B, one of whose roots is A, and Vieta&#039;s formulas are used to determine the other root to the quadratic.However, I do not fully see the need to take the minimal solution, can someone explain that step for me?

Ismael Wilkinson · Accepted Answer

Step 1The transformation exists for all solutions, as do some others such as exchanging a and b, or changing their signs. Call two solutions equivalent if they can be reached from each other by sequences of those transformations.Step 2The purpose of choosing a &quot;minimal&quot; solution is to pick out a simplest equivalent of any given solution, which might then be seen to have some other special property, such as   a  b  =  0, that has implications for all its equivalent solutions (such as all of them having   (      a    2    +      b    2    )      /    (  a  b  +  1  ) a perfect square, since that ratio is invariant under the transformations).

The minimal solution (A, B) with respect to some function of A and B, usually A + B, is taken. The equation is then rearranged into a quadratic with coefficients in terms of B, one of whose roots is A, and Vieta's formulas are used to determine the other root to the quadratic.

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