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Statement

Equation (2) (p. 136). For every $\xi$,

$$(9\xi^4)^3+(3\xi-9\xi^4)^3+(1-9\xi^3)^3=1. \qquad(2)$$

The paper concludes that each integer $\xi$ gives a non-trivial integer solution of $x^3+y^3+z^3=1$ (its equation (1)), the trivial ones being those such as $x=1$, $y=-z$; so (1) has infinitely many non-trivial integer solutions, which answers the question Mordell had put to Mahler by letter.

The paper says "for every integer $\xi$"; at $\xi=0$ the identity gives the trivial solution $(0,0,1)$, every $\xi\ne0$ gives a non-trivial one, and distinct values of $\xi$ give distinct solutions, since $z=1-9\xi^3$ determines $\xi$ (an observation of this page, not of the paper).

The companion identities (p. 137). The paper continues with

$$(9d^3\xi^4)^3+(3d\xi-9d^3\xi^4)^3+d(1-9d^2\xi^3)^3=d, \qquad(3)$$

and concludes that $x^3+y^3+dz^3=d$ and $d^2(x^3+y^3)+z^3=1$ both have infinitely many integer solutions; and with

$$(6d^2\xi^3+1)^3+(-6d^2\xi^3+1)^3+d(-6d\xi^2)^3=2, \qquad(4)$$

identically in $\xi$, that there are infinitely many integers $x,y,z$ with $x^3+y^3+dz^3=2$, or with $d^2(x^3+y^3)+z^3=d^2$. The paper then calls this last identity a special case of a family (5) of identities in $n\ge3$ variables, which its unnumbered theorem on p. 138 (theorem_p138_general) uses.

The second form from (3) follows by writing $x=dx'$, $y=dy'$, since the first two terms of (3) are divisible by $d$. The same step does not give the second form printed after (4): writing $Z=dz$ in $x^3+y^3+dz^3=2$ gives $d^2(x^3+y^3)+Z^3=2d^2$, not $d^2$ (an observation of this page, not of the paper).

Source. K. Mahler, Note on Hypothesis K of Hardy and Littlewood, J. London Math. Soc. 11 (1936), no. 2, 136-138: equation (2) on p. 136, equations (3), (4) and (5) on p. 137. The edition read is identified on the source card.

Read depth. Claims checked: the identities were read on the printed pages, and (2), (3), (4) and the homogenized form (2') were checked by direct expansion at small integer values. Nothing here is independently reviewed.

Proof pointer

Page 136. The paper specializes a classical parametrization of $x^3+y^3+z^3=u^3$ by quartic forms in $f,g,f',g'$, built from the quadratic forms $\rho,\rho',\sigma,\sigma'$ (cited to Dickson's History of the theory of numbers, vol. 2, p. 555) to $u=1$ by taking $f'=1$, $g'=0$, $f=3g$, and then puts $2g=\xi$. Identity (2) can also be checked by expanding both sides.

Dependencies

None in this corpus; the parametrization of $x^3+y^3+z^3=u^3$ is cited to Dickson.

Bears on

• Problem 322: through its homogenized form (2'), identity (2) is the input to the theorem on p. 138 (theorem_p138_cubes), which gives many representations of twelfth powers as sums of three cubes. Identity (2) alone concerns signed cubes summing to $1$ and gives no representation count of the kind the problem asks about.