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Proof Problem

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

Please help me with the problem, Thanks!

Note by Swapnil Das
2 years, 2 months ago

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

We can have:

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

or

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

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

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

Satyajit Mohanty - 2 years, 2 months ago

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Solution master: @Satyajit Mohanty

Mehul Arora - 2 years, 2 months ago

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

Mehul Arora - 2 years, 2 months ago

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

Dev Sharma - 2 years, 2 months ago

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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 - 2 years, 2 months ago

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