Number Theory or Algebra? Part 6

Well, if $p=3$ then $p$ , $p+2$ and $p+4$ are all primes. So, $p=3$ works.

Now $p$ has to be an odd integer, otherwise none of $p$ , $p+2$ and $p+4$ are primes.

If $p$ is a positive odd integer greater than $3$ , then either it is of the form $(3k+1)$ , or it is of the form $(3k-1)$ where $k$ is a positive integer.

If $p=3k+1$ , then $p+2$ is divisible by $3$ .

If $p=3k-1$ , then $p+4$ is divisible by $3$ .

So, it is impossible for $p$ , $p+2$ , $p+4$ to be all primes if $p>3$ .

And our final answer is $\boxed{3}$ [the sum of one integer is the integer itself.

For more information, you could take a look at the first statement of this problem.

Python code:

def checkprime(n):

s=0

for i in range(n/2):

    if n%(i+2)==0:

        s=s+1

if s==0:

    return 0

else:

    return 1

def sum(k):

q=0

for p in range(k):

    m=p+2

    a=checkprime(m)+checkprime(m+2)+checkprime(m+4)

    if a==0:

        q=q+m


return q

$if\quad we\quad consider\quad any\quad prime\quad p\quad then\quad there\quad can\quad be\quad 3\quad cases\quad p\quad \equiv \quad 1\quad (mod\quad 3)\\ p\quad \equiv \quad 2\quad (mod\quad 3)\quad or\quad 3|p\quad 3\quad would\quad divide\quad p\quad only\quad if\quad p\quad =\quad 3\\ trying\quad out\quad for\quad other\quad cases\quad :\\ case\quad 1\\ if\quad p\quad \equiv \quad 1\quad (mod\quad 3)\quad then\quad p\quad +\quad 2\quad \equiv \quad 3\quad (mod\quad 3)\quad which\quad means\quad 3\quad |\quad p\quad +\quad 2\\ but\quad it\quad can't\quad be\quad possible\quad as\quad p\quad +\quad 2\quad is\quad a\quad prime.\\ case\quad 2\\ if\quad p\quad \equiv \quad 2\quad (mod\quad 3)\quad then\quad p\quad +\quad 4\quad \equiv \quad 6\quad (mod\quad 3)\quad which\quad means\quad 3\quad |\quad p\quad +\quad 4\\ but\quad it\quad can't\quad be\quad possible\quad as\quad p\quad +\quad 2\quad is\quad a\quad prime.\\ finally\quad we\quad find\quad 3\quad |\quad p\quad and\quad p\quad is\quad a\quad prime\quad as\quad well\quad so\quad p\quad =\quad 3$

except p=3 no other value of p exist which makes p ,p+1 p+2 prime