Abstract
SummaryDudeney's puzzles of a hundred years ago included writing integers (specifically 9 and 17) as sums of two cubes of positive rational numbers (where in the former case, a solution other than 1, 2 is required). We study the corresponding equations x3 + y3 = 9 and x3 + y3 = 17 as examples of specific elliptic curves. The group structure is introduced, and the smallest solutions found for Dudeney's puzzles. Generalization to x3 + y3 = n reveals that sometimes the smallest rational solution can be very large, for example when n = 94 and n = 4981: the latter solution involves fractions with numerator and denominator having almost 17 million digits.
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