Let $p$ and $q$ be two prime numbers less than $900$ billion. If $p + 6, p + 10, q + 4, q + 10$ and $p + q + 1$ are all primes, what is the greatest value that $p + q$ can take?

Can you help me to solve this problem (useful relations exist but I don't know about them)?

  • $\begingroup$ Take it as a generalization of "$p,p+3$ are both primes" (which implies $p=2$) $\endgroup$
    – reuns
    Oct 3 '17 at 8:12
  • $\begingroup$ @reuns How did you get $+3$ in there? $\endgroup$
    – Arthur
    Oct 3 '17 at 8:13

$p+10$ is prime, so not a multiple of $3$, so $p$ can't be $1$ less than a multiple of $3$. Similarly $q$ can't be $1$ less than a multiple of $3$. $p$ and $q$ can't both be $1$ more than a multiple of $3$, since then $p+q+1$ would be a multiple of $3$. So one of them is a multiple of $3$, and since prime must be equal to $3$.

In fact $q$ must be $3$ since $p+6$ is prime. Now the conditions reduce to $p+4$ ($=p+q+1$), $p+6$ and $p+10$ must all be prime. There are lots of possible choices for $p$; you just need to find the largest such $p$ in the range given.

  • $\begingroup$ Note that since $p+4, p+6$ is a pair of twin primes, you can probably find a list to check them. $\endgroup$
    – Arthur
    Oct 3 '17 at 8:14
  • $\begingroup$ Yes, probably, or it doesn't take too long to search by computer, starting at 900 billion and working down. $\endgroup$ Oct 3 '17 at 8:25
  • 1
    $\begingroup$ oeis.org/A258088 is pertinent here (but doesn't go far enough to provide an answer). $\endgroup$ Oct 3 '17 at 9:21

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