To solve 1729 as a sum of two cubes using natural numbers, you need to find two numbers whose cubes add up to 1729. These numbers can be positive or negative, as long as they are both natural numbers.
The two cubes that add up to 1729 are 1^3 and 12^3. This can be written as 1^3 + 12^3 = 1729.
The numbers 1 and 12 were discovered by mathematician Srinivasa Ramanujan, who famously stated that 1729 is the smallest number expressible as a sum of two cubes in two different ways. This means that there are only two possible combinations of cubes that add up to 1729, and 1 and 12 are one of them.
No, 1729 can only be expressed as a sum of two cubes in two different ways: 1^3 + 12^3 and 9^3 + 10^3. This has been proven by mathematicians and is known as the "taxicab number" or "Hardy-Ramanujan number".
Finding sums of cubes for certain numbers can be helpful in number theory and cryptography. It can also be used to solve problems in different fields of science, such as physics and chemistry. Additionally, it showcases the beauty and patterns in mathematics and can inspire new discoveries and breakthroughs.