Adding 2's is always a prime?

Number Theory Level 3

If p , p + 2 , p + 4 p, p+ 2, p + 4 are all prime numbers, can p + 6 p + 6 also be a prime number?

No Yes

Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Steven Perkins
Jul 17, 2017

One of the numbers p, p+2, and p+4 must be divisible by 3. If they are all primes then one of them must be 3.

The only solution is thus 3, 5, 7.

p+6 is therefore 9 which is NOT prime.

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One of the numbers p, p+2, and p+4 must be divisible by 3.

Why is this true? Why can't one of these numbers be divisible by another number, say 2017?

Pi Han Goh - 3 years, 10 months ago

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I left that as an exercise for the student. :-)

They may be divisible by some other prime (if and only if they ARE that prime), but that isn't important to the answer.

It's not hard to see if you look at these numbers modulo 3. They are equivalent to p, p+2, p+1.

Clearly (to me anyway) one of these is equivalent to 0 (mod 3), and thus is divisible by 3.

Steven Perkins - 3 years, 10 months ago

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