This question is inspired by the announcement of a proof that "fake twin" primes, i.e. pairs of consecutive primes differing by at most K, are -in infinite number- where K is a fixed integer which can be taken = 70,000,000.
It is not known OTOH whether the above would hold where K would be = 2 instead (the twin prime conjecture).
Anyway, my question may be (hopefully is) much simpler : it's been known after Viggo Brun that the sum of reciprocals of actual "twin primes" is finite (whether the number of terms in this sum is finite or infinite). Does the sum of reciprocals of the "fake twins" as defined above (for K=70 million) also converge ? Does it follow from an easy extension of Brun's method ?
I should make it clear I am no expert, just curious... I do not have accesss to a proof of Brun's original theorem, even a sketchy one.