# A Note on the Diophantine Equation x 3 + y 3 + z 3 = 3

@article{Cassels1985ANO, title={A Note on the Diophantine Equation x 3 + y 3 + z 3 = 3}, author={John W. Cassels}, journal={Mathematics of Computation}, year={1985}, volume={44}, pages={265} }

Any integral solution of the title equation has x =y z (9). The report of Scarowsky and Boyarsky [3] that an extensive computer search has failed to turn up any further integral solutions of the title equation prompts me to give the proof of a result which I noted many years ago and which might be of use in further work (cf. footnote on p. 505 of [2]). THEOREM. Any integral solution of (1) X3 + y3 + z3 = 3 has (2) x -y -z (9). Proof. Trivially, (3) x -y --z -1 (3). We work in the ring Z[p] of… Expand

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#### References

SHOWING 1-2 OF 2 REFERENCES

Mordell, "Integer solutions of x2 + y2 + z2 + 2xyz = n,

- J. London Math. Soc,
- 1953

Nachtrag zum cubischen Reciprocitätssatze_"

- Reine Angew. Math., v. 28,
- 1844