- On May 14, 1927, the author presented to the Mathematical Association of America a paper on the case r = 2, a = 1 of equation (1) below.
- If n = r, (2) may not have a solution. For example, 1/(x1x2)+1/(x2x1)=3/5 has no solution, as can be quickly shown by the method of trial and error.
- D. R. Curtiss, On Kellogg's Diophantine problem, this Monthly, vol. 29 (1922), pp. 380–387. He found the maximum value that xn can assume in the equation l/x1+ l/x2+…+ l/xn=1 when x1, x2,…, xn-1 are positive integers and x1 ≦ x2 ≦… ≦ xn and showed how to extend his theory so as to obtain a similar result for the equation 1/x1+1/x2+…+1/xn=b/[(m+1)b-1] when b, m are positive integers and n > r. Tanzo Takenouchi, On an indeterminate equation, Proceedings of the Physico-Mathematical Society of Japan, vol. 3, No. 6, pp. 78–92. He obtained the results just mentioned by a method which is entirely different from that of Curtiss.
- This definition is somewhat like one which Takenouchi used in his paper referred to in footnote 3.
- Cf. footnote 3.
- See p. 81 of Takenouchi's article referred to in footnote 3.
The American Mathematical Monthly
Volume 36, 1929 - Issue 3