E428 -- Observationes circa bina biquadrata, quorum summam in duo alia biquadrata
resolvere liceat
(Observations about two biquadratics, of which the sum is able to be resolved into two other biquadratics)
Summary:
(based on Jordan Bell's abstract)
Euler considers solutions to A4 + B4 = C4 + D4 and gives a method for finding
solutions. Using this method he finds the solutions A = 2219449, B = -555617, C = 1584749, D = 2061283 and
A = 477069, B = 8497, C = 310319, D=428397; the first four numbers satisfy the equation, but the second four do not. He
also states the "Euler quartic conjecture," which says that there is no biquadratic that is the sum of three other biquadratics.
According
to the records, it was presented to the St. Petersburg Academy on January 13, 1772.
Publication:
-
Originally published in Novi Commentarii academiae scientiarum Petropolitanae 17, 1773, pp. 64-69
-
Opera Omnia: Series 1, Volume 3, pp. 211 - 217
- Reprinted in Commentat. arithm. 1, 1849, pp. 473-476 [E428a]
Documents Available:
- Original publication: E428
- English translation (Jordan Bell): E428
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E428 include:
- Bremner A., “On Euler's quartic surface.” Mathematica Scandinavica, 61 (2), pp. 165-180 (1987).
- Zajta AJ., “Solutions of the Diophantine equation A4 + B4 = C4 + D4.” Mathematics of Computation, 41 (164), pp. 635-659 (1983).
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