Answer:
(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 <em>remainder</em>.
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.
<em>First case</em>: . Then 
that can be prime, and 
that is not prime. Hence, there are not such three primes with .
<em>Second case</em>: . 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.
