On the diophantine equation $x^5+y^5=z^5+t^5$

Question

It is known that the diophantine equation $x^5+y^5=z^5+t^5$, has infinity many trivial solutions. Has this equation been proven to have other integer solutions?

Answer 1

I. The OP asks for "...other integer solutions". We will interpret this to allow for Gaussian integers $a+bi$, an important class of quadratic integers. Hence, a non-trivial solution is,

$$(a+ci)^5+(b-ci)^5 = (a-ci)^5+(b+ci)^5$$

where $a^2+b^2 = 2c^2,\,$ and $a\neq b.\,$ A version was first found by Desboves. For example, $(a,b,c) = (1,7,5)$, so,

$$(1+5i)^5+(7-5i)^5 = (1-5i)^5+(7+5i)^5$$

and infinitely more.

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II. However, if the OP wishes for plain vanilla integers, then no non-trivial solutions are known to,

$$a^5+b^5 = c^5+d^5 = N$$

where $N < 1.02\times10^{26}$ (Guy, 1994) cited in Mathworld. However, this has been updated to $N < 4.01\times10^{30}$ (Ekl, 1998). This roughly implies there are no solutions with both $(a,b)<10^6.$

P.S. Ekl's 25-year-old bound seems to need an update, though the Lander-Parkin-Selfridge conjecture assumes there are none anyway.

See https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture