How many natural numbers are there with the property that they can be expressed as sum of cubes of two natural numbers in two different ways.

probably infinitely many, but the smallest is

1729 = 1^3 + 12^3 = 9^3 + 10^3

as G.H. Hardy learned from Ramanujan.

steve now give me two different ways for it for my last question

To find the number of natural numbers that can be expressed as the sum of cubes of two natural numbers in two different ways, let's break down the problem step by step:

1. First, we need to find the range of natural numbers. Since there is no upper limit mentioned, we assume it to be infinite (i.e., there are infinitely many natural numbers).

2. Next, we'll express the natural numbers as the sum of cubes of two natural numbers. Let's assume the natural number is "n".

3. The sum of cubes of two natural numbers can be expressed as a^3 + b^3, where a and b are natural numbers.

4. Now, we need to find the possible pairs (a, b) where a^3 + b^3 = n. We can use nested loops to iterate through all possible pairs of values for a and b. The loops should run from 1 to the square root of n (as b = n - a^3, and both a and b must be natural numbers).

5. For each pair of (a, b), we calculate the sum of cubes as a^3 + b^3. If it matches the given natural number "n", increment a counter variable by 1.

6. Repeat steps 4-5 until we have checked all pairs of (a, b) for the natural number "n".

7. Finally, repeat steps 2-6 for all natural numbers starting from 1 until we reach a natural number for which the counter value is greater than or equal to 2. The moment we find a natural number with a counter value equal to or greater than 2, it means the number can be expressed as the sum of cubes of two natural numbers in two different ways.

By following these steps, we can find and count the natural numbers that have the property of being expressed as the sum of cubes of two natural numbers in two different ways.