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The number 1729 is famously the smallest positive integer expressible as the sum of two positive cubes in two different ways ($1729=1^3+12^3=9^3+10^3$). There is plenty of work on "taxicab numbers" - the smallest sums of cubes in $n$ different ways (which always exist) - Here's Ivars Peterson at MAA And here's another detailed analysis. (Does anyone know anything about the "Bill Butler" referred to in the second article)

However the sequence which caught my attention is OEIS A016078 - 4, 50, 1729, 635318657 which gives the smallest numbers which are sums of positive $n^{th}$ powers in two ways. Is there any more recent work or prospect of identifying such numbers for fifth powers and above? And should they be named as in the title of this post?

[This question arises from a much more frivolous one, which was closed, in which I learned why a $50^{th}$ birthday was special in this particular way].

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    http://math.stackexchange.com/questions/226333/generalised-hardy-ramanujan-numbers2017-03-08

2 Answers 2

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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.

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    @draks..., in the Erdos question at F30 in Guy, $a$ and $b$ are meant to be whole numbers. I'm not aware of any recent progress.2013-11-27
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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.