Number Theory or Algebra? Part 7

The only $n$ that will work is $4$ .

The proof that no other values of $n$ exist can be proved in the same way as the solution by Mursalin Habib for this problem . Just change into $5k$ , $5k + 1$ , ..., $5k + 5$

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+1

    a=checkprime(m+1)+checkprime(m+3)+checkprime(m+7)+checkprime(m+9)+checkprime(m+13)+checkprime(m+15)

    if a==0:

        q=q+m

return q

when have to put a value of such that n+1 is even and satisfy all n+3,n+7,n+9 and n+15 all are odd then only 4 satisfy above requirement and no other value except 4 satisfy above requirements