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
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Sources

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