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A survey is presented of the more important solution methods of the equation of the title. When space permits, a brief description of the methods and numerical examples are also given. The paper concludes with an incomplete list of 218 primitive nontrivial solutions in rational integers not exceeding 106.

Solutions of the Diophantine equation 𝐴⁴+𝐵⁴=𝐶⁴+𝐷⁴