Help me Part 3

If $a +b +c =0$ and $a, b, c$ are integers, can you prove or disprove that $\frac{a^4+b^4+c^4}{2}$ is a perfect square?

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`2 \times 3`	$2 \times 3$
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Comments

With $c = -(a + b)$ we have that

$\dfrac{a^{4} + b^{4} + c^{4}}{2} = \dfrac{a^{4} + b^{4} + (a + b)^{4}}{2} = \dfrac{a^{4} + b^{4} + a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}}{2} = a^{4} + 2a^{3}b + 3a^{2}b^{2} + 2ab^{3} + b^{4} = (a^{2} + ab + b^{2})^{2}$

thus proving that the given expression is a perfect square when $a + b + c = 0$ .

Brian Charlesworth - 2 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$