Are there Integers which are Sum of 2 Fifth Powers in 2 Ways?

Open Question

It is not known whether there exists an integer which equals the sum of two fifth powers in $2$ different ways.

That is, it is not known whether the Diophantine equation:

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

has a solution.

Progress

It is known that no such solutions exist for sums up to $1 \cdotp 02 \times 10^{26}$.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,375,298,099$
• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1,375,298,099$