- #1

Math100

- 756

- 205

- Homework Statement
- If ## p ## and ## p^{2}+8 ## are both prime numbers, prove that ## p^{3}+4 ## is also prime.

- Relevant Equations
- None.

Proof:

Suppose ## p ## and ## p^{2}+8 ## are both prime numbers.

Since ## p^{2}+8 ## is prime, it follows that ## p ## is odd, so ## p\neq 2 ##.

Let ## p>3 ##.

Then ## p^{2}\equiv 1 \mod 3 ##,

so ## p^{2}+8\equiv 0 \mod 3 ##.

Note that ## p^{2}+8 ## can only be prime for ## p=3 ##.

Thus ## p^{3}+4=27+4=31 ##, which is also prime.

Therefore, if ## p ## and ## p^{2}+8 ## are both prime numbers,

then ## p^{3}+4 ## is also prime.

Suppose ## p ## and ## p^{2}+8 ## are both prime numbers.

Since ## p^{2}+8 ## is prime, it follows that ## p ## is odd, so ## p\neq 2 ##.

Let ## p>3 ##.

Then ## p^{2}\equiv 1 \mod 3 ##,

so ## p^{2}+8\equiv 0 \mod 3 ##.

Note that ## p^{2}+8 ## can only be prime for ## p=3 ##.

Thus ## p^{3}+4=27+4=31 ##, which is also prime.

Therefore, if ## p ## and ## p^{2}+8 ## are both prime numbers,

then ## p^{3}+4 ## is also prime.

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