Prove Integer Solution for $x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=k$

[anemone]

Prove that the equation $x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=k$ has an integer solution for any integer $k$.

 

[anemone]

Hint:

Spoiler

Note that $k=6n$ can be represented as a sum of four cubes.

 

[magneto1]

Spoiler

The hint follows from:
\[
(x+1)^3 + (x-1)^3 + (-x)^3 + (-x)^3 = 6x.
\]
Moroever, we have that under $\pmod 6$,
$(\pm 1)^3 \equiv \pm1$, $(\pm 2)^3 \equiv \pm 2$,
$3^3 \equiv 3$. We can choose any of the two of the cubes to form
$6x + k$ where $k = 1..5$.

 

[anemone]

Spoiler

The hint follows from:
\[
(x+1)^3 + (x-1)^3 + (-x)^3 + (-x)^3 = 6x.
\]
Moroever, we have that under $\pmod 6$,
$(\pm 1)^3 \equiv \pm1$, $(\pm 2)^3 \equiv \pm 2$,
$3^3 \equiv 3$. We can choose any of the two of the cubes to form
$6x + k$ where $k = 1..5$.

You're right magneto, well done and thanks for participating!:)

 

[CellCoree]

I would approach this problem by first acknowledging that this is a well-known mathematical problem known as the "sum of five cubes." It has been extensively studied by mathematicians over the years, and various proofs have been proposed to show that it has integer solutions for all values of $k$.

One approach to proving the existence of integer solutions for this equation is through the use of modular arithmetic. By considering the equation modulo different numbers, we can show that there are certain patterns and constraints that the solutions must follow. This can help us narrow down the possible values for $x_1, x_2, x_3, x_4,$ and $x_5$ and eventually find an integer solution for any given $k$.

Another approach is through the use of Diophantine equations, which are equations that involve only integer solutions. By transforming the given equation into a Diophantine equation and using techniques such as descent, we can show that there are integer solutions for any value of $k$.

Additionally, there are other methods such as using identities and properties of cubic equations that can help in proving the existence of integer solutions for this equation.

In conclusion, as a scientist, I can confidently say that the equation $x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=k$ has an integer solution for any integer $k$, as it has been proven and studied extensively by mathematicians using various techniques and methods.