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Does there exist a positive integer $n>5$ such that the sum of the two largest primes less than $n$ equals $n$? If yes, lovely! If not, what is the largest prime gap possible between the two largest primes less than $n$?

The first question could also be rewritten as $2*p - n = d$

where $p$ is the largest prime less than $n$ and $d$ is the difference between $p$ and the second largest prime less than $n$.

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    $\begingroup$ Maybe, Bertrand's postulate helps here. It states that for every $n>1$ , $(n,2n]$ contains at least one prime. $\endgroup$
    – Peter
    Commented Jul 18, 2023 at 18:01
  • $\begingroup$ The Wiki page for Bertrand’s postulate mentions a 1952 theorem of Nagura stating that for $n \geq 25$, there is a prime between $n$ and $6n/5$. In particular, for $n \geq 36$, there are two primes between $25/36n$ and $n$, so the sum of the two largest primes less than $n$ is at least $(5/6+25/36-1)n=19n/36 > n/2$. It’s easy to check what happens with $n<36$. I don’t understand your second question, though. $\endgroup$
    – Aphelli
    Commented Jul 18, 2023 at 20:38
  • $\begingroup$ There is no limit for the prime gap between the two largest prime numbers less than $n$ because you can make the gap between consecutive primes $p<q$ arbitarily large and just choose $n:=q+1$. $\endgroup$
    – Peter
    Commented Jul 21, 2023 at 0:27
  • $\begingroup$ @Peter I understand that, but I am looking for an 'n' such that the gap between p & q is greater than or equal to n/2. $\endgroup$
    – hefe
    Commented Jul 25, 2023 at 18:45

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