Problems & Puzzles: Puzzles

Problem 77.  Accumulating primes in a 2-pans balance (A332788) Ivan N. Ianakiev posted the following nice puzzle: I think I have an interesting question that is also on OEIS website (sequence A332788), but which no one has  answered yet: Take a double-pan balance scale and name the pans "negative" and "positive". At each step, the question is: "Is there an unused prime that would balance the scale if added to the positive pan? If the answer is yes, add that prime to the positive pan. Otherwise, add the smallest unused prime to the negative pan. Q.  Is the number of primes in the positive pan infinite?

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On November 16, 2020, Adam Stinchcombe wrote:

I am reminded of the G.H.Hardy quote:  “it is easy to make clever guesses.”  I am not optimistic of being able to solve something like this.

A sufficient condition which I am currently inclined to believe runs (all variables are positive integers and the pj are the ordered and enumerated primes): for all m, for all n, for all i>n, there is a k such that M + sum(pj,j=i..i+k) is prime.  Then M is the balance in the negative pan, i is where you are at in the list of primes.  If p(i+k) is taken already then just restart with a new M’ = M + sum(pj,j=i..i+k) and find a new k’.  As counter-evidence, the order of the primes is important: if you order the primes starting with 2,3 and then alternate +1 and -1 mod 6, then starting with M=4 the sum is never a prime, alternating between  0 mod 2 and 0 mod 3.

In the process of thinking about this problem, I had a thought that every infinite subset of the integers with pairwise relatively prime numbers has a subset that adds to a prime, but I came up with a way to construct a set such that every subset sums to a composite.

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