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A Closed Form for Representing Integers as Sums and Differences of Cubes
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Michael Nyblom

School of Science

RMIT University

Melbourne, Victoria 3001

Australia

**Abstract:**

By applying an existing characterization for a positive integer to be
represented as a sum of two cubes of positive integers, we construct
an elementary proof of Ramanujan's famous result—namely, that
the number 1729 is the smallest positive integer represented as a sum
of two cubes in two different ways. Similarly, by applying an existing
characterization for a positive integer to be represented as a difference
of cubes of two positive integers, we also apply this characterization
to construct the generating function for a sequence of integer ordered
pairs (*a*_{n}, *b*_{n}) ≠
(*a'*_{n}, *b'*_{n})
satisfying *a*_{n}^{3}
+ *b*_{n}^{3} =
*a'*^{3}_{n} + *b'*^{3}_{n},
which are distinct from Ramanujan's "near integer" solutions
to Fermat's equation—namely, those satisfying
*a*_{n}^{3}
+ *b*_{n}^{3} =
*a*_{n}^{3} +
(–1)^{n}.

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(Concerned with sequences
A001235
A004999
A011541
A343708.)

Received December 24 2021; revised versions received January 2 2022; April 8 2022.
Published in *Journal of Integer Sequences*,
April 10 2022.

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