Twin primes (a) let p > 3 be a prime. prove that p is of the form 3k +1 or 3k – 1 for some integer k. (b) twin primes are pairs of prime numbers p and q that have a difference of 2. use part (a) to prove that 5 is the only prime number that takes part in two different twin prime pairs.

(a) Let us recall the division algorithm: given two positive integers and there exist other two positive integers and such that

where and is called the remainder.

So, given any positive integer and 3 we can write

where . Thus, every can be written as

Now, notice that . Hence, every number can be written as , or or .

A number is prime if and only if its only factors are 1 and itself. So, a number of the form cannot be prime. Therefore, every primer number is of the form or .

(b) Assume that there are three prime numbers such that , and are prime.

By the previous exercise or . Let us analyze both cases separately.

First case: . Then that can be prime, and that is not prime. Hence, there are not such three primes with .

Second case: . Then, that can be prime, and that cannot be prime. Hence, there are not such three primes with .

Therefore, there are no three primes  of the form , and , except for 3, 5 and 7.

Notice that this is only possible because 5=2*3-1 and 2*3-3=3, that is the only ‘‘multiple’’ of 3 that is prime.

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