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

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?

• Such numbers are called taxibac numbers , for exponent $5$ or more , no such taxicab number is known. Commented Dec 5, 2023 at 23:05
• euh , NO , no solutions are known apart from $x=z,y=t$.
– mick
Commented Dec 5, 2023 at 23:09
• mathworld.wolfram.com/DiophantineEquation5thPowers.html
– mick
Commented Dec 5, 2023 at 23:13
• @mick The case $p=5$ is solvable by Gaussian integers, in reference to your question on $p=11$ and Eisenstein integers. Commented Dec 6, 2023 at 14:47
• @TitoPiezasIII I am somewhat aware of that. I do not know the parametrization of all solutions if there is one. But I know there are solutions. Not so surprising since solutions for $p=5$ might even have ordinary integer solutions( that is an open problem) and many arguments exist for it, and those arguments get stronger for a bigger ring ofc.
– mick
Commented Dec 6, 2023 at 19:25

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.

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.

• In what year did Desboves find that ?
– mick
Commented Dec 6, 2023 at 19:32
• @mick Adolphe Desboves lived during the 1800s, so it was within that time. Commented Dec 7, 2023 at 4:01
• yes I was aware of the century but not the exact date of that result. Thanks anyway.
– mick
Commented Dec 7, 2023 at 12:01