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Source. Theorem 12, p. 462, with its proof on pp. 462--463, of Kurt Mahler, On the lattice points on curves of genus 1, Proc. London Math. Soc. (2) 39 (1935), 431--466, the edition named on the source card.

Read depth. Claims checked: the statement was read clause by clause on p. 462, the proof for its structure. Nothing here is independently reviewed.

Statement

Theorem 12 (p. 462), quoted: "Suppose that $f(x)$ is a polynomial of exact degree 3 or 4 in $x$ with rational coefficients, and that $t\ge1$ is an integer. Then there is an integer $k\ne0$ such that, for at least $t$ different rational values of $x$, the polynomial $kf(x)$ is the square of an integer."

The paper notes (p. 462) that Theorem 10 is a special case. Section 27 (p. 463) applies it to obtain Theorems 13 and 14: for every $t\ge1$ there is a polynomial $a_0x^2+a_1$ ($a_0a_1\ne0$) with integer coefficients that is a perfect cube, respectively a perfect fourth power, for at least $t$ different integer values of $x$.

Proof pointer

Pp. 462--463. The case of a multiple root is called trivial; otherwise $y^2=f(x)$ is a curve of genus 1 with an elliptic uniformization, and a parabola $y=Ax^2+Bx+C$ meets it in four points whose elliptic arguments sum to a constant. Osculating parabolas give the points with arguments $u_1,-3u_1,9u_1,\ldots,(-3)^{t-1}u_1$, chosen distinct and finite on a curve $y^2=k_1f(x)$ through a suitable rational point; clearing denominators then gives the integer $k$.

Dependencies

None outside the paper's own method.

Bears on

No Erdős problem in the corpus.