Proof Problem

Find all the integral solutions of ${ x }^{ 4 }+{ y }^{ 4 }+{ z }^{ 4 }-{ w }^{ 4 }=1995$ .

Please help me with the problem, Thanks!

#NumberTheory

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Comments

We know that $a^4 \equiv 0,1 \pmod {16} \quad \forall \ a \in \mathbb Z$ .

We can have:

$x^4 + y^4 + z^4 - w^4 \equiv -1,0,1,2,3 \pmod {16}$

$x^4 + y^4 + z^4 - w^4 \equiv 0,1,2,3,15 \pmod {16}$

But $1995 \equiv 11 \pmod {16}$ . So no integral solutions exist!

@Swapnil Das - I'd say it's a Number Theory Problem!

Satyajit Mohanty - 5 years, 10 months ago

Yeah! It is! Great solution @Satyajit Mohanty :)

Mehul Arora - 5 years, 10 months ago

Solution master: @Satyajit Mohanty

Mehul Arora - 5 years, 10 months ago

why did you work with mod16? why not other numbers? is there any rule?

Dev Sharma - 5 years, 10 months ago

I worked with mod 16 because fourth powers have an interesting property as they are equivalent to only 0 or 1 modulo 16. I don't have any idea about other modulos for 16th powers. Well, one can try further.

Satyajit Mohanty - 5 years, 10 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}`	$\boxed{123}$