Unsolved Problem 18:

Are there distinct positive integers, a, b, c, and, d such that a^5+b^5=c^5+d^5?

It is known that 1^3+12^3=9^3+10^3 and 133^4+134^4=59^4+158^4, but no similar relation is known for fifth powers. Other remarkable identities are 27^5+84^5+110^5+133^5=144^5 and 2682440^4+15365639^4+18796760^4=20615673^4.


[Guy 1994]
Richard K. Guy, Unsolved Problems in Number Theory, second edition. Springer Verlag. New York: 1994. Page 140.
Diophantine Equation of Degree 5
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