Perfect Squares Only

Number Theory Level 5

Let \(a,b,c\) be nonzero perfect squares such that \(b = a+c\). The equation \(ax^2 - bx + c = 0\) has the following properties:

  • It has a real root \(d\) that is greater than 1. (The other root may or may not be real.)
  • The square root of the quadratic discriminant is a prime number.

Among all such \((a,b,c,d)\), pick the one such that the discriminant is minimum. Find \(\sqrt{abcd}\).


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