Numbers not Expressible as Sum of Fewer than 19 Fourth Powers/Historical Note
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Historical Note on Numbers not Expressible as Sum of Fewer than 19 Fourth Powers
It was noted by Leonard Eugene Dickson that there are no other positive integers less than $4100$ needing $19$ fourth powers to express them.
This limit was reported by David Wells in his Curious and Interesting Numbers of $1986$.
Jean-Marc Deshouillers, François Hennecart and Bernard Landreau extended this to $10^{245}$ in $2000$.
In $2005$, Koichi Kawada, Trevor Dion Wooley and Jean-Marc Deshouillers showed that the sequence is complete beyond $10^{220}$.
Sources
- 1933: L.E. Dickson: Recent progress on Waring's theorem and its generalizations (Bull. Amer. Math. Soc. Vol. 39: pp. 701 – 727)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $559$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $559$
- 2000: J.-M. Deshouillers, François Hennecart and Bernard Landreau: Waring's Problem for Sixteen Biquadrates -- Numerical Results (Journal de théorie des nombres de Bordeaux Vol. 12: pp. 411 – 422)
- 2005: J.-M. Deshouillers, K. Kawada and T.D. Wooley: On sums of sixteen biquadrates (Mem. Soc. Math. Fr. Vol. 100: p. 120)