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Given $S$ of positive integers $>$ $1$ is there some combination with even $SUM$ > $2$ that is NOT the sum of two primes?

$SUM$ = 10

$S$ = $[4,6]$

$No$,

Sum of Two Primes $5 + 5 = 10$.

Combination $4+6=10$

It has an $O(1)$ algorithm if Goldbach is True (always output $NO$). Otherwise, it would seem to be NP-complete because it would require solving Subset-Sum.

Question

Will a many-one-reduction from $Subset-sum$ work for this decision problem in poly-time?

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  • $\begingroup$ It's a very special instance of subset sum. I don't see how you'd reduce general subset sum to this particular case. $\endgroup$ – Yuval Filmus Jun 21 at 21:17
  • $\begingroup$ @YuvalFilmus As a function problem (output subset-sum with integers) is virtually the same for the general subset-sum with the exception that the answer is always NO if the conjecture is true. $\endgroup$ – Dingle Berry Jun 21 at 22:50
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If there is a finite number of exceptions to the Goldbach conjecture then it is still solvable in linear time.

Note that there is a non-zero probability that the Goldbach conjecture is false. Heuristically, the probability for an infinite number of exceptions is zero.

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  • $\begingroup$ Mathematically, has it been proven to be an infinite number of exceptions to the Goldbach Conjecture? (In the sense, $IF$ proven false)? Anyway, thanks for the answer! $\endgroup$ – Dingle Berry Jun 23 at 1:52

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