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Activity Discussion Math A Math trivia

• # A Math trivia

Posted by on June 24, 2023 at 8:24 pm

Which is the only number which is the sum of cubes of two different pairs? (It has a special name too)

replied 1 year, 1 month ago 4 Members · 3 Replies
• 3 Replies
• ### Krati

Member
June 24, 2023 at 9:11 pm
0

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>
</sup>

• ### Supratik

Member
June 24, 2023 at 10:41 pm
0

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:

1. 1729 = 1Â³ + 12Â³
2. 1729 = 9Â³ + 10Â³

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 ðŸ™‚

• ### Shishank

Member
June 25, 2023 at 11:44 am