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

  title={A Note on the Diophantine Equation x 3 + y 3 + z 3 = 3},
  author={John W. Cassels},
  journal={Mathematics of Computation},
  • J. Cassels
  • Published 1985
  • Mathematics
  • Mathematics of Computation
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… 
8 Citations
90.34 On triples of integers having the same sum and the same product
  • Juan Plá
  • Mathematics
    The Mathematical Gazette
  • 2006
x Sx + mx P = 0, where m is an arbitrary complex number. For each m, d, e and/ will be the roots of this equation. But the problem starts to be more interesting and difficult if we wish to have (a,
Primitive Representations of Integers by $x^3+y^3+2z^3$
A well-known open problem is to show that the cubic form $x^3+y^3+2z^3$ represents all integers. An obvious variant of this problem is whether every integer can be {\em primitively} represented by
The density of zeros of forms for which weak approximation fails
The weak approximation principal fails for the forms x + y + z = kw, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these
New integer representations as the sum of three cubes
A new algorithm is described for finding integer solutions to x 3 + y 3 + z 3 = k for specific values of k and this is used to find representations forvalues of k for which no solution was previously known, including k = 30 and k = 52.
Groupe de Brauer et points entiers de deux surfaces cubiques affines
Il est connu depuis Ryley [Ryl25] que tout entier, et meme tout nombre rationnel, peut s’ecrire comme somme de trois cubes de nombres rationnels. La question de savoir quels entiers s’ecrivent comme
Cracking the problem with 33
  • A. Booker
  • Mathematics
    Research in Number Theory
  • 2019
Inspired by the Numberphile video "The uncracked problem with 33" by Tim Browning and Brady Haran, we investigate solutions to $x^3+y^3+z^3=k$ for a few small values of $k$. We find the first known
On a question of Mordell
This paper concludes a 65-y search with an affirmative answer to Mordell’s question and strongly supports a related conjecture of Heath-Brown and makes several improvements to methods for finding integer solutions to x3+y3+z3=k for small values of k.


Mordell, "Integer solutions of x2 + y2 + z2 + 2xyz = n,
  • J. London Math. Soc,
  • 1953
Nachtrag zum cubischen Reciprocitätssatze_"
  • Reine Angew. Math., v. 28,
  • 1844