Monday Math 34

Today, we have another bit of number theory: what are the positive integers n such that n3+1 is prime?

Well, we recall that a3+b3=(a+b)(a2ab+b2). Thus, n3+1=(n+1)(n2n+1).
Now, for n3+1 to be prime, one of those factors must be equal to one. For positive n, n+1>1, and so we have n2n+1=1, which means
n2n=0
n(n-1)=0
which, for positive n, gives only n=1, for n3+1=13+1=2 prime. All other positive n give composite numbers.

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