1729 is considered a magic number because it is the smallest positive integer that can be expressed as the sum of two cubes in two different ways.
The Sum of Two Cubes
This fascinating property makes 1729 unique in the world of numbers. Let's break it down:
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First combination: 1729 can be expressed as 10³ + 9³
- 10³ = 10 10 10 = 1000
- 9³ = 9 9 9 = 729
- 1000 + 729 = 1729
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Second combination: 1729 can also be expressed as 12³ + 1³
- 12³ = 12 12 12 = 1728
- 1³ = 1 1 1 = 1
- 1728 + 1 = 1729
The Significance of 1729
The number 1729 is famously known as the Hardy-Ramanujan number after the mathematician G.H. Hardy visited Srinivasa Ramanujan in hospital. Hardy remarked that the number 1729 on the taxi he arrived in was dull. Ramanujan immediately responded that it was far from dull being the smallest number expressible as a sum of two cubes in two different ways.
Table of Sums of Cubes
To illustrate this further, here's a simple table showing the two unique ways to arrive at 1729:
| Expression | Calculation | Result |
|---|---|---|
| 10³ + 9³ | 1000 + 729 | 1729 |
| 12³ + 1³ | 1728 + 1 | 1729 |
This unique feature, where a single number can be derived from two different sets of cubed values, is what gives 1729 its magical distinction in mathematics.
