New years radical!!(in advance)

Algebra Level 5

Find the sum of all positive integers (a,b)(a,b) such that

(a+b+a2+b2)(ab+a2+b2)(a+ba2+b2)4(a+b+a2+b2)(ab+a2+b2)(a+ba2+b2)4=4(a+b+a2+b2).\small{\small{\sqrt{\dfrac{(-a+b+\sqrt{a^2+b^2})(a-b+\sqrt{a^2+b^2})(a+b-\sqrt{a^2+b^2})}{4}\sqrt{\dfrac{(-a+b+\sqrt{a^2+b^2})(a-b+\sqrt{a^2+b^2})(a+b-\sqrt{a^2+b^2})}{4}\sqrt{\dots\dots}}}=4(a+b+\sqrt{a^2+b^2})}}.

Details and Assumptions

  • (a,b)(a,b) is same as (b,a)(b,a), no need to add twice.
  • This is inspired by Mursalin Habib.
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