Smallest Positive Integer which is Sum of 2 Fourth Powers in 2 Ways

Theorem

The smallest positive integer which can be expressed as the sum of $2$ fourth powers in $2$ different ways is:

\(\ds 635 \, 318 \, 657\)

\(=\)

\(\ds 59^4 + 158^4\)

\(\ds \)

\(=\)

\(\ds 133^4 + 134^4\)

Proof

The fact that these are the smallest can be demonstrated by calculation.

$\blacksquare$

Historical Note

This question was raised by Leonhard Paul Euler in $1772$.

He it was who found the smallest solution, which he did in $1802$.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $59$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $635,318,657$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $59$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $635,318,657$

• Piezas, Tito III and Weisstein, Eric W. "Diophantine Equation--4th Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquation4thPowers.html