Are there infinite many prime-"tuples", for which $p_{n+1}+p_{n+2}−{p_n}=p_{n+3}$

Question

I found this interesting sequence for primes:

$2+3=5 $
$7+11-5 = 13$
17 + 19 - 13 = 23
29 + 31 - 23 = 37
41 + 43 - 37 = 47

Unfornutaley, now the pattern breaks, but comes back:

primes = []
c = 0
for possiblePrime in range(2, 20000):

# Assume number is prime until shown it is not.
isPrime = True
for num in range(2, possiblePrime):
    if possiblePrime % num == 0:
        isPrime = False
        break
if isPrime:
    primes.append(possiblePrime)
for i in range(5, 1000, 3):
  a = primes[i-2] + primes[i-1]
  a = a - primes[i-3]
  if a != primes[i]:
    c = c+1
    print (""+repr(primes[i-2]) + " + " + repr(primes[i-1]) +" - "+ 
    repr(primes[i-3]) + " != " + repr(primes[i]))
  else:
    print("" + repr(primes[i - 2]) + " + " + repr(primes[i - 1]) + " - " + repr(primes[i - 3]) + " = " + repr(
        primes[i]))

Are the infinitifly many of these tuples, for which $p_{n+1}+p_{n+2}−{p_n}=p_{n+3}$?

Kind regards

Answer 1

Your question, I believe, is whether or not there are an infinite number of values of $n$ where

$$p_{n + 1} + p_{n + 2} - p_n = p_{n + 3} \Rightarrow p_{n + 1} - p_n = p_{n + 3} - p_{n + 2} \tag{1}\label{eq1}$$

where $n \in \mathbb{N}$ and $p_n$ is the $n$'th prime. Let

$$g_n = p_{n + 1} - p_n \tag{2}\label{eq2}$$

where $g_n$ is the $n$'th prime gap. Then \eqref{eq1} becomes

$$g_n = g_{n + 2} \tag{3}\label{eq3}$$

In other words, the prime gap between $2$ primes is the same as that for $2$ primes higher. You can see this, for example, in your $7 + 11 − 5 = 13$ equation where $7 - 5 = 13 - 11 = 2$.

The prime gaps, on average, do increase as primes get larger, but it is a very slow increase (the Prime Number Theorem shows the average gap at $m$ is about $\log m$). Also, the prime gap values fluctuate a lot, with there not being any particular pattern that anybody has ever found, apart from very basic ones like they are all even except for the first one of $3 - 2 = 1$.

Although I am quite certain nobody has proven that \eqref{eq3} is true for an infinite number of $n$, nor do I think it would be easy to do, I believe it's true. Otherwise, it would be a rather unusual property of primes for this relation to only hold for a finite number of times.

Answer 2

It is believed that there are infinitely many $n$ such that $n$, $n+2$, $n+6$, and $n+8$ are all prime. If that's true, it solves your problem in the affirmative. But no one knows how to prove the belief is justified.