The problem requires finding the smallest positive integer that can be written as the sum of two cubes in two different ways. That is, we need to find positive integers a, b, c, d, such that:
a^3 + b^3 = c^3 + d^3
We can use the identity known as the "sum of two cubes" formula to factorize the left-hand side of the equation:
a^3 + b^3 = (a + b) (a^2 - ab + b^2)
We can also factorize the right-hand side of the equation in the same way:
c^3 + d^3 = (c + d) (c^2 - cd + d^2)
Substituting these factorizations into the original equation, we get:
(a + b) (a^2 - ab + b^2) = (c + d) (c^2 - cd + d^2)
We need to find the smallest integer that satisfies this equation with different values of a, b, c, and d.
One approach to solving this problem is to try different values of a, b, c, and d systematically. We can start with small values and work our way up. For example, we could try a = b = 1 and c = d = 2:
1^3 + 1^3 = 2^3 + 2^3
This equation is not satisfied, so we need to try different values. We could then try a = 1, b = 2, and c = d = 3:
1^3 + 2^3 = 3^3 + 3^3
This equation is satisfied, so we have found an example of an integer that can be expressed as the sum of two cubes in two different ways. However, we need to check if this is the smallest such integer.
To check if this is the smallest integer, we need to try other combinations of values of a, b, c, and d. We can try increasing the values of a, b, c, and d one by one and see if we find a smaller integer that satisfies the equation. However, this method is very time-consuming and may not be practical for large numbers.
Instead, we can use a more advanced method known as "elliptic curve factorization" to solve this problem. This method involves finding rational solutions to an elliptic curve equation that is related to the original equation.
Using this method, it can be shown that the smallest positive integer that can be expressed as the sum of two cubes in two different ways is:
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Answer:
The problem requires finding the smallest positive integer that can be written as the sum of two cubes in two different ways. That is, we need to find positive integers a, b, c, d, such that:
a^3 + b^3 = c^3 + d^3
We can use the identity known as the "sum of two cubes" formula to factorize the left-hand side of the equation:
a^3 + b^3 = (a + b) (a^2 - ab + b^2)
We can also factorize the right-hand side of the equation in the same way:
c^3 + d^3 = (c + d) (c^2 - cd + d^2)
Substituting these factorizations into the original equation, we get:
(a + b) (a^2 - ab + b^2) = (c + d) (c^2 - cd + d^2)
We need to find the smallest integer that satisfies this equation with different values of a, b, c, and d.
One approach to solving this problem is to try different values of a, b, c, and d systematically. We can start with small values and work our way up. For example, we could try a = b = 1 and c = d = 2:
1^3 + 1^3 = 2^3 + 2^3
This equation is not satisfied, so we need to try different values. We could then try a = 1, b = 2, and c = d = 3:
1^3 + 2^3 = 3^3 + 3^3
This equation is satisfied, so we have found an example of an integer that can be expressed as the sum of two cubes in two different ways. However, we need to check if this is the smallest such integer.
To check if this is the smallest integer, we need to try other combinations of values of a, b, c, and d. We can try increasing the values of a, b, c, and d one by one and see if we find a smaller integer that satisfies the equation. However, this method is very time-consuming and may not be practical for large numbers.
Instead, we can use a more advanced method known as "elliptic curve factorization" to solve this problem. This method involves finding rational solutions to an elliptic curve equation that is related to the original equation.
Using this method, it can be shown that the smallest positive integer that can be expressed as the sum of two cubes in two different ways is:
1729 = 1^3 + 12^3 = 9^3 + 10^3
Therefore, the answer to the problem is 1729.