The number of positive integers with the property that they cann be expressed as the sum of the cubes of 2 positive integers in two different ways is

Options

A. 1
B. 100
C. infinite
D. 0

Explanation

To solve the problem of finding the number of positive integers that can be expressed as the sum of the cubes of two positive integers in two different ways, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find positive integers \( n \) such that there exist positive integers \( a, b, c, d \) satisfying:
\[ n = a^3 + b^3 = c^3 + d^3 \]
where \( (a, b) \) and \( (c, d) \) are different pairs of positive integers.

2. **Identifying Known Results**: There are known examples of such integers. A famous example is \( 1729 \), known as the Hardy-Ramanujan number, which can be expressed as:
\[ 1729 = 1^3 + 12^3 = 9^3 + 10^3 \]
This means \( 1729 \) can be expressed as the sum of cubes in two different ways.

3. **Finding More Examples**: We can look for other numbers that can be expressed as the sum of two cubes in different ways. The next number after \( 1729 \) that satisfies this condition is \( 4104 \):
\[ 4104 = 2^3 + 16^3 = 9^3 + 15^3 \]

4. **Continuing the Search**: Continuing this process, we find that \( 13832 \) and \( 20683 \) also satisfy the condition:
\[ 13832 = 2^3 + 24^3 = 18^3 + 20^3 \]
\[ 20683 = 10^3 + 27^3 = 19^3 + 24^3 \]

5. **Conclusion**: The known positive integers that can be expressed as the sum of the cubes of two positive integers in two different ways are \( 1729, 4104, 13832, 20683 \), and potentially more. However, there are infinitely many pairs of positive integers, and thus infinitely many such sums.

6. **Final Count**: Since we can find infinitely many such integers, we conclude that the answer is that there are infinitely many positive integers that can be expressed as the sum of the cubes of two positive integers in two different ways.

### Final Answer:
The number of positive integers that can be expressed as the sum of the cubes of two positive integers in two different ways is **infinite**.