Problems & Puzzles:
Accumulating primes in a 2-pans balance
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
number of primes in the positive pan infinite?
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.