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A Math trivia
Posted by Sayak on June 24, 2023 at 8:24 pmWhich is the only number which is the sum of cubes of two different pairs? (It has a special name too)
Shishank replied 3 months ago 4 Members · 3 Replies 
3 Replies

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Given a large number, <code datanotranslation=””>n, find all positive numbers less than or equal to <code datanotranslation=””>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 datanotranslation=””>m <= n that can be expressed as:
<code datanotranslation=””>m = (a<sup>3</sup> + b<sup>3</sup>) = (c<sup>3</sup> + d<sup>3</sup>) for distinct <code datanotranslation=””>a, <code datanotranslation=””>b, <code datanotranslation=””>c, <code datanotranslation=””>d.
For example,
If <code datanotranslation=””>n = 25000, <code datanotranslation=””>m can be any of <code datanotranslation=””>1729, <code datanotranslation=””>4104, <code datanotranslation=””>13832, or <code datanotranslation=””>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 HardyRamanujan 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 HardyRamanujan 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 HardyRamanujan 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 HardyRamanujan Number.