30-Apr-1995
## 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.
## Reference:

- [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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