Numbers not Expressible as Sum of Fewer than 19 Fourth Powers

From ProofWiki
Jump to navigation Jump to search

Theorem

The following positive integer are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers:

$79, 159, 239, 319, 399, 479, 559$

This sequence is A046050 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

On a case-by-case basis:

From Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers:

$79 = 15 \times 1^4 + 4 \times 2^4$


From 159 is not Expressible as Sum of Fewer than 19 Fourth Powers:

$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$


From 239 is not Expressible as Sum of Fewer than 19 Fourth Powers:

$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$


From 319 is not Expressible as Sum of Fewer than 19 Fourth Powers

$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$

or:

$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$


From 399 is not Expressible as Sum of Fewer than 19 Fourth Powers

$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$

or:

$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$


From 479 is not Expressible as Sum of Fewer than 19 Fourth Powers

$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$

or:

$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$


From 559 is not Expressible as Sum of Fewer than 19 Fourth Powers

$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$

or:

$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$



Also see


Historical Note

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, Fran├žois 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