A Little Number Theory to Begin the School Year

Prove that if p and p²+8 are prime then so is p³+4.

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A Little Number Theory to Begin the School Year Solution

Case 1: If p is 3 then p²+8=17 and p³+4=31, so it is true in this case.

Case 2: If p>3 then either p=3k+1 or p=3k-1 for some integer, k.

If p=3k+1 then p²+8 = 9k²+6k+9, which is a multiple of 3 and therefore not prime.

If p=3k-1 then p²+8 = 9k²-6k+9, which is also not prime.

So the statement is true (vacuously) in this case as well.

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