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.

Staff - 2 years, 8 months ago

This is from the First HK TST.

- 2 years, 6 months ago

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 ?

- 2 years, 8 months ago

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

Staff - 2 years, 8 months ago

And for $$6k \pm 1$$ ?

- 2 years, 8 months ago

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

- 2 years, 6 months ago