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Hi, I was wondering if anyone could provide a solution for the following question:

Find all positive integers n for which n+ n + 1 is a prime number

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David W. | Experienced ProfExperienced Prof
4.4 4.4 (42 lesson ratings) (42)
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      n = 1;   (n8 + n + 1) = 3              [that's all of them]

It often helps to reword the question. Let's use this logic:
     If A then B
        means
     If not B then not A

So,
      If (n8+n+1) is not a prime number, then n is not a positive integer.

Def of prime number: a positive integer whose only integer factors are 1 and itself.

        Note: for n=1, (n8+n+1)=3, which is a prime number.

Now, we must show that (n8+n+1) can be factored and that the factors cannot be 1 and (n8+n+1), except for n=1.

Yes, (n8+n+1) can be factored:
             (n8 + n + 1) = (n2 + n + 1)*(n6 – n5 + n3 – n2 + 1)

Now, to determine that (n8+n+) is not prime, we must show that these factors are never both 1 and the number (n8+n+1) itself, except for n=1.

For positive integers n, the factor (n2+n+1) is never equal to 1 and it is never equal to (n8+n+1), except when n=1, since that would mean that n2=n8.

Therefore (n8+n+1) is not a prime number (except for n=1).

q.e.d.