Olympiad Adventure

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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Note to self: 1 is not a prime number.

Calvin Lin Staff - 2 years, 8 months ago

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This is from the First HK TST.

Sualeh Asif - 2 years, 6 months ago

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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 ?

Sachin Vishwakarma - 2 years, 8 months ago

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

Calvin Lin Staff - 2 years, 8 months ago

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

Sachin Vishwakarma - 2 years, 8 months ago

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@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

Sualeh Asif - 2 years, 6 months ago

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