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Activity Discussion Math A Math trivia Reply To: A Math trivia

• ### Krati

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June 24, 2023 at 9:11 pm
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Given a large number, <code data-no-translation=””>n, find all positive numbers less than or equal to <code data-no-translation=””>n that can be represented as the sum of two cubes for at least two different pairs.

In other words, find all positive numbers <code data-no-translation=””>m <= n that can be expressed as:

<code data-no-translation=””>m = (a<sup>3</sup> + b<sup>3</sup>) = (c<sup>3</sup> + d<sup>3</sup>) for distinct <code data-no-translation=””>a, <code data-no-translation=””>b, <code data-no-translation=””>c, <code data-no-translation=””>d.

For example,

If <code data-no-translation=””>n = 25000, <code data-no-translation=””>m can be any of <code data-no-translation=””>1729, <code data-no-translation=””>4104, <code data-no-translation=””>13832, or <code data-no-translation=””>20683 as these numbers can be represented as the sum of two cubes for two different pairs.

1729 = 1<sup>3</sup> + 12<sup>3</sup> = 9<sup>3</sup> + 10<sup>3</sup>
4104 = 2<sup>3</sup> + 16<sup>3</sup> = 9<sup>3</sup> + 15<sup>3</sup>
13832 = 2<sup>3</sup> + 24<sup>3</sup> = 18<sup>3</sup> + 20<sup>3</sup>
20683 = 10<sup>3</sup> + 27<sup>3</sup> = 19<sup>3</sup> + 24<sup>3</sup>

1729, the Hardy-Ramanujan Number, is the smallest number which can be expressed as the sum of two different cubes in two different ways.<sup>
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