Sum of Two Rational 4th Powers but not Two Integer 4th Powers
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Theorem
$5906$ is the smallest integer which can be expressed as the sum of two rational $4$th powers, but not two integer $4$th powers.
Proof
- $5906 = \paren {\dfrac {149} {17} }^4 + \paren {\dfrac {25} {17} }^4$
Suppose $5906$ is a sum of two integer $4$th powers.
We have:
- $9^4 = 6561 > 5906$
which shows that no $4$th power greater than $8^4$ is in the sum.
- $7^4 + 7^4 = 4802 < 5906$
which shows that some $4$th power greater than $7^4$ is in the sum.
So the sum must contain $8^4$.
We have:
- $5906 - 8^4 = 1810$
but $1810$ is not an integer $4$th power.
Therefore $5906$ is not a sum of two integer $4$th powers.
This needs considerable tedious hard slog to complete it. In particular: It remains to show that it is the smallest such. It is, acccording to A NEW CHARACTERIZATION OF THE INTEGER 5906 by A. Bremner and P. Morton 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 {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1983: Andrew Bremner and Patrick Morton: A new characterization of the integer 5906 (Manuscripta Mathematica Vol. 44: pp. 187 – 229)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5906$