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 Eisenstein integers, where p is a cube root of unity. If a E Z[p] is prime to then there is precisely one unit e = +pi (j = 0,1,2) such that ea 1 (3). The supplement [1] to the law of cubic reciprocity states that if so E Z[p] is prime, s1 (3), then 3 is a cubic residue of so in Z[p] precisely when v a (9) for some a E Z. It follows that if a E Z[p], a 1 (3) and if 3 is congruent to a cube modulo a, then a b (9) for some b E Z. Put a = -p 2X-py, so a = x + (x y)p 1 (3) by (3). By (1) we have z3 3 (a), so the preceding remarks apply. Hence x y 0 (9). Finally, (2) follows by symmetry. Department of Pure Mathematics and Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge CB2 1SB, England Received March 28, 1984. 1980 Mathematics Subject Classification. Primary 1OB15. ?01985 American Mathematical Society 0025-5718/85 $1.00 + $.25 per page 265 This content downloaded from 207.46.13.115 on Sat, 08 Oct 2016 04:45:07 UTC All use subject to http://about.jstor.org/terms 266 J. W. S. CASSELS 1. G. EISENSTEIN, Nachtrag zum cubischen Reciprocitatssatze.... J. Reine Angew. Math., v. 28, 1844, pp. 28-35. 2. L. J. MORDELL, Integer solutions of x2 + y2 + z2 + 2xyz = n, J. London Math. Soc., v. 28, 1953, pp. 500-510. 3. M. SCAROWSKY & A. BOYARSKY, A note on the Diophantine equation x' + yf + zn = 3, Math. Comp., v. 42, 1984, pp. 235-236. This content downloaded from 207.46.13.115 on Sat, 08 Oct 2016 04:45:07 UTC All use subject to http://about.jstor.org/terms