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Find all natural number \(n\) such that \(n\), \(n^2 + 10\), \(n^2 - 2\), \(n^3 + 6\), \(n^5 + 36\) are all prime numbers.

Note by Dev Sharma 2 years, 8 months ago

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2 \times 3

2^{34}

a_{i-1}

\frac{2}{3}

\sqrt{2}

\sum_{i=1}^3

\sin \theta

\boxed{123}

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Note to self: 1 is not a prime number.

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This is from the First HK TST.

One question from me: Is every odd prime representable in form of 4k \( \pm\) 1, where k \( \epsilon \) N ? Or Is every prime greater than 3 is representable in form of 6k \( \pm\) 1, where k \( \epsilon \) N ?

Every odd number (not necessarily prime) is representable in the form \( 4k \pm 1 \) for a suitable k.

And for \( 6k \pm 1 \) ?

@Sachin Vishwakarma – Yes every prime is representable in the form \(6k\pm 1\).

Proof: \(6k, 6k +2,6k+4\) are divisible by 2

\(6k+3\) is divisible by 3

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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

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TopNewestNote to self: 1 is not a prime number.

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This is from the First HK TST.

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One question from me: Is every odd prime representable in form of 4k \( \pm\) 1, where k \( \epsilon \) N ? Or Is every prime greater than 3 is representable in form of 6k \( \pm\) 1, where k \( \epsilon \) N ?

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Every odd number (not necessarily prime) is representable in the form \( 4k \pm 1 \) for a suitable k.

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And for \( 6k \pm 1 \) ?

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Proof: \(6k, 6k +2,6k+4\) are divisible by 2

\(6k+3\) is divisible by 3

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