The Diophantine equation x³ + y³ + z³ = k has known solutions for certain values of k. However, Fermat's Last Theorem implies that there are no solutions for k = 2, highlighting the profound nature of the theorem.
The equation x³ + y³ + z³ = k represents a Diophantine equation, and it is related to the concept of sums of three cubes. This problem is associated with the famous Fermat's Last Theorem, which states that there are no three positive integers x, y, and z such that xⁿ + yⁿ = zⁿ for any integer value of n greater than 2.
For the specific case of cubes, it is known that there are solutions for certain values of k. Here are some examples:
1. k = 1:
x = 1, y = 1, z = 1
2. k = 2:
No known solution (Fermat's Last Theorem implies that there are no solutions for k = 2)
3. k = 3:
x = 1, y = 1, z = 1
4. k = 4:
x = 2, y = -1, z = -1
5. k = 5:
No known solution
6. k = 6:
x = 2, y = 1, z = -1
Complete question:
Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 6.