Numbers not Expressible as Sum of Fewer than 19 Fourth Powers

From ProofWiki
Jump to navigation Jump to search


It has been suggested that this page or section be merged into Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Mergeto}} from the code.


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:

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



This theorem requires a proof.
In particular: It needs to be shown that there are no other numbers known with this property.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


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, 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

Retrieved from "https://proofwiki.org/w/index.php?title=Numbers_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers&oldid=524512"