Anning 1945 integral distances

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construction_p598: Anning and Erdős's construction of a set of points dense on a circle with all pairwise distances rational, followed by their record of Ulam's question whether a dense set in the plane can have all distances rational, which they state they cannot answer.

theorem_p598: Anning and Erdős's theorem that for every n there are n points in the plane, not all on a line, with all mutual distances integers, while no infinite set of points in the plane, not all on a line, has all mutual distances integers.

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N. H. Anning, P. Erdős: Integral distances, Bull. Amer. Math. Soc. 51 (1945), 598--600 (MR 7,164a; Zentralblatt 63, Index). The copy read for this card is the archive scan at https://users.renyi.hu/~p_erdos/1945-01.pdf, and none of its three pages prints a copyright or license line; the publisher's article page redirected to the current Bulletin volume rather than the article (https://www.ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/, read 2026-10-02), and the publisher's copyright policy page states "AMS permits the noncommercial use of its copyrighted works for educational purposes only, such as to quote brief passages or to copy small portions of content for personal use in teaching or research" and names Creative Commons licenses only for its open-access series, not the Bulletin (https://www.ams.org/publications/authors/ctp, read 2026-10-02), every other right reserved.

The paper proves two complementary facts: for every n there is a set of n points in the plane, not all on a line, whose pairwise distances are all integers, but there is no infinite such set. The construction (p. 598) takes the circle x^2 + y^2 = 1/4, uses primes p_i = 4k+1 with p_i^2 = a_i^2 + b_i^2 (a_i, b_i nonzero) to place points at distance b_i/p_i from (-1/2, 0), and applies Ptolemy's theorem to four concyclic points inductively to make every pairwise distance rational; enlarging the radius clears denominators. A footnote records that Anning had given 24 points on a circle with integral distances (Amer. Math. Monthly 22 (1915), p. 321). The authors state that the points of this construction are very likely dense on the circle and that they cannot prove it, and they give (pp. 598--599) a set dense on the circle with all distances rational: with P_1 = (-1/2, 0), P_2 = (1/2, 0), X_1 the point of the circle at distance 3/5 from P_1 and alpha the angle P_2P_1X_1 (so sin alpha = 4/5 and cos alpha = 3/5), which they call known to be an irrational multiple of pi, the points X_i of the circle with angle P_1P_2X_i equal to i alpha. A second configuration (p. 599) takes an odd m^2 with d divisors, the d solutions of m^2 = x_i^2 - y_i^2, and the points (m, 0) and (0, y_i). The authors then record Ulam's question whether a dense set in the plane can have all distances rational, and state that they do not know the answer. The impossibility proof (pp. 599--600) has two steps: no line contains infinitely many of the points, since integer distances give d(PQ_j) <= d(PQ_i) + d(Q_iQ_j) - 1 for a point P off the line, which a perpendicular-foot estimate rules out for large distances; and the points near a limiting direction lie within bounded distance of a line, so three far-apart non-collinear ones give the same contradiction. The last paragraph states without proof that a similar argument rules out infinitely many points in n-dimensional space, not all on a line, with all distances integral.

Source: https://users.renyi.hu/~p_erdos/1945-01.pdf.

Read status. Claims checked: the Theorem (p. 598), the dense construction and Ulam's question (pp. 598--599) were read clause by clause on the page images; the proofs were read in full and followed in outline, not checked.

Bears on. #213: the Theorem shows that no infinite plane set with no three points on a line has all distances integers; the finite sets of its proof lie on a circle (p. 598) and those of the second configuration all but one on a line (p. 599), so for n >= 4 neither meets the problem's conditions, and it settles no instance. #130: by the Theorem the problem's integer-distance graph on an infinite set with no three points on a line and no four on a circle has no complete subgraph on infinitely many vertices; it bounds neither the size of finite complete subgraphs nor the chromatic number. #212: the paper records the problem's question from Ulam (p. 599) and states that the authors do not know the answer; its rational-distance set is dense on a circle, not in the plane.

Results. the Theorem (p. 598, proof pp. 598--600); the dense rational-distance set on a circle and Ulam's question (pp. 598--599).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.