Observations about two biquadratics, of which the sum is able to be resolved into two other biquadratics
Euler considers solutions to A4+B4 = C4+D4 and gives a method for finding solutions. Using this method, he finds the solutions (A, B, C, D) = (2219449, -555617, 1584749, 2061283) and (477069, 8497, 310319, 428397); the first set of numbers satisfies the equation, but the second does not. He also states the "Euler quartic conjecture," which claims that there is no biquadratic that is equal to the sum of three other biquadratics. (Based on Jordan Bell's translation abstract.)
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 17, pp. 64-69.
Opera Omnia Citation
Series 1, Volume 3, pp.211-217.