This paper (E716) was published in Nova acta Academiae scientiarum imperialis petropolitanae, Volume 13 (1795/96), pp. 45-63. It was also included in Commentationes Arithmeticae, Volume II, as Number LXVIII, pp. 281-293 (E791). Euler starts with Fermat's Last Theorem and mentions the proofs for the cases n=3 and n=4 which he had completed himself earlier. He then moves on to make the sum of powers conjecture, which was later disproved in the second half of the 20th century. In this context he discusses his discovery of 134^4+133^4=158^4+59^4, which he calls unexpected. Euler derives the title equation from A^4+B^4=C^4+D^4, generalizing it to some extent, and derives three methods by which special solutions may be obtained. He gives two different sets of explicit formulas for the case m=n=1. Euler shows that each solution of ab(aa+bb)=cd(cc+dd) leads to a conjugate solution. Euler considers the problem of making two particular binary forms simultaneously square, and shows how its solution is connected to the case m=n=1.

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