[seqfan] Infinite partial sumsets in the primes

[Fred Lunnon]

>From Terry Tao's latest "What's New" at

https://terrytao.wordpress.com/2023/01/26/infinite-partial-sumsets-in-the-primes/

--- might there be interesting (if difficult) `earliest' sequences to
investigate here?
WFL

*Theorem 1*

   - (i) If the Hardy-Littlewood prime tuples conjecture
   <https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture>
    (or the weaker conjecture of Dickson
   <https://en.wikipedia.org/wiki/Dickson%27s_conjecture>) is true, then
   there exists an increasing sequence [image: {b_1 < b_2 < b_3 < \dots}] of
   primes such that [image: {b_i + b_j + 1}] is prime for all [image: {i <
   j}].
   - (ii) Unconditionally, there exist increasing sequences [image: {a_1 <
   a_2 < \dots}] and [image: {b_1 < b_2 < \dots}] of natural numbers such
   that [image: {a_i + b_j}] is prime for all [image: {i<j}].
   - (iii) These conclusions fail if “prime” is replaced by “positive
   density subset of the primes” (even if the density is equal to 1).