Answer:
Explanation:
A Diophantine equation is an equation that is only satisfied by integer solutions.
To solve the equation x³+y³+z³=k, we need to find integer values of x, y, and z that will make the equation true for a given value of k.
One approach to solving this equation is to try different values for x, y, and z and see if they work. For example, we can try setting x = 1, y = 1, and z = 1, which gives us 1³+1³+1³ = 3. We can then try setting x = 2, y = 2, and z = 2, which gives us 2³+2³+2³ = 12.
Another approach is to use the identity (a+b+c)³ = a³+b³+c³+3(a+b)(b+c)(c+a). This identity allows us to express the left side of the equation as a sum of cubes plus a multiple of the product of three pairs of sums. This can be useful if we are able to find values for a, b, and c that make the product on the right side equal to k.
For example, if we set a = 1, b = 2, and c = 3, we get (1+2+3)³ = 1³+2³+3³+3(1+2)(2+3)(3+1) = 36 = 1+8+27+3(3)(5)(4) = 36. This shows that the equation x³+y³+z³=36 has the solution x = 1, y = 2, and z = 3.
We can use this approach to find solutions for other values of k as well. For example, if we set a = 4, b = 5, and c = 6, we get (4+5+6)³ = 4³+5³+6³+3(4+5)(5+6)(6+4) = 4³+5³+6³+3(9)(11)(10) = 216 = 64+125+216+990 = 216. This shows that the equation x³+y³+z³=216 has the solution x = 4, y = 5, and z = 6.
We can continue this process to find solutions for other values of k. However, it is important to note that in general, it may not be possible to find integer solutions for all values of k. In some cases, there may be no solutions, or there may be an infinite number of solutions.