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A Math trivia
Which is the only number which is the sum of cubes of two different pairs? (It has a special name too)
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3 Replies
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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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The number you are referring to is known as “1729,” which is also called the Hardy-Ramanujan number. It is the smallest positive integer that can be expressed as the sum of two cubes in two different ways. The two pairs of cubes that add up to 1729 are:
- 1729 = 1³ + 12³
- 1729 = 9³ + 10³
This number gained significance when the mathematician G.H. Hardy visited his colleague Srinivasa Ramanujan in the hospital. Hardy mentioned that he traveled in a taxi with the number 1729, considering it to be a rather dull and uninteresting number. However, Ramanujan disagreed and stated that it was a very interesting number since it was the smallest number that could be expressed as the sum of two cubes in multiple ways. This incident led to the number 1729 being named the Hardy-Ramanujan number in their honor.
Hope it helps 🙂
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1729, the is the smallest number which can be expressed as the sum of two different cubes in two different ways. 1729 is the sum of the cubes of 10 and 9 – cube of 10 is 1000 and cube of 9 is 729; adding the two numbers results in 1729. It is known as the Hardy-Ramanujan Number.
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