Fourth Powers which are Sum of 4 Fourth Powers
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Theorem
The following positive integers are such that their fourth powers can be expressed as the sum of the fourth powers of $4$ other positive integers with no common factors:
- $353, 651, 2487, 2501, 2829, \ldots$
This sequence is A039664 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
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Examples
353
- $353^4 = 30^4 + 120^4 + 272^4 + 315^4$
651
- $651^4 = 240^4 + 340^4 + 430^4 + 599^4$
Sources
- July 1973: Kermit Rose and Simcha Brudno: More About Four Biquadrates Equal One Biquadrate (Math. Comp. Vol. 27, no. 123: pp. 491 – 494) www.jstor.org/stable/2005655
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $651$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $353$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $651$
- Piezas, Tito III and Weisstein, Eric W. "Diophantine Equation--4th Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquation4thPowers.html