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.