Generalised Hardy-Ramanujan Numbers
Solution 1:
Guy, Unsolved Problems In Number Theory, 3rd edition, D1, writes, "... it is not known if there is any nontrivial solution of $a^5+b^5=c^5+d^5$. Dick Lehmer once thought that there might be a solution with a sum of about 25 decimal digits, but a search by Blair Kelly yielded no nontrivial solution with sum $\le1.02\times10^{26}$."
At F30, Guy writes, "... $x^5$ is a likely answer to the following unsolved problem of Erdos. Find a polynomial $P(x)$ such that all the sums $P(a)+P(b)$ ($0\le a\lt b$) are distinct."
The book was published in 2004. I don't know whether there has been any progress since.
Solution 2:
It is necessary to solve the equation:
$$x^5+y^5+z^5=q^5$$
For integers complex numbers solutions exist. $j=\sqrt{-1}$
Making this change.
$$a=p^2-2ps-s^2$$
$$b=p^2+2ps-s^2$$
$$c=p^2+s^2$$
You can write the solution.
$$x=jc+b$$
$$y=jc-b$$
$$z=a-jc$$
$$q=a+jc$$
$p,s$ - integers.