New user? Sign up

Existing user? Log in

Prove that n^4+4 is a composite number for n>1?

Note by Naitik Sanghavi 3 years, 1 month ago

Easy Math Editor

*italics*

_italics_

**bold**

__bold__

- bulleted- list

1. numbered2. list

paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

> This is a quote

This is a quote

# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

2 \times 3

2^{34}

a_{i-1}

\frac{2}{3}

\sqrt{2}

\sum_{i=1}^3

\sin \theta

\boxed{123}

Sort by:

Note that \(n^{4} + 4 = (n^{4} + 4n^{2} + 4) - 4n^{2} = (n^{2} + 2)^{2} - (2n)^{2} = ((n^{2} + 2) - 2n)((n^{2} + 2) + 2n).\)

For \(n \gt 1\) both of these last two bracketed terms are \(\gt 1,\) thus proving that \(n^{4} + 4\) is composite for \(n \gt 1.\)

Log in to reply

Perfect!!!..

Sophie-Germain Identity.

\(a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)\)

\((a,b)=(n,1)\) gives \(n^4+4=(n^2+2+2n)(n^2+2-2n)\),

which is composite for \(n\ge 2\), since \(n^2+2-2n\ge 2\).

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestNote that \(n^{4} + 4 = (n^{4} + 4n^{2} + 4) - 4n^{2} = (n^{2} + 2)^{2} - (2n)^{2} = ((n^{2} + 2) - 2n)((n^{2} + 2) + 2n).\)

For \(n \gt 1\) both of these last two bracketed terms are \(\gt 1,\) thus proving that \(n^{4} + 4\) is composite for \(n \gt 1.\)

Log in to reply

Perfect!!!..

Log in to reply

Sophie-Germain Identity.

\(a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)\)

\((a,b)=(n,1)\) gives \(n^4+4=(n^2+2+2n)(n^2+2-2n)\),

which is composite for \(n\ge 2\), since \(n^2+2-2n\ge 2\).

Log in to reply