A Goldbug number is an even number 2m for which there exists some subset of the prime non-divisors (PNDs) of 2m, 2 < p1 < p2 < p3 < ... < pk < m, such that (2m-p1)*(2m-p2)*(2m-p3)*...*(2m-pk) has only p1,p2,p3,...,pk as factors and one of the pi is between n/2 and n for even n and (n+1)/2 and n-1 for odd n. We do not need to consider the case where n is prime since then n itself is a Goldbach Pair.
Goldbug's Algorithm to find Goldbach Pairs for Non-Goldbug Numbers
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1. By Lemma 1 we need only consider prime non-divisors (PNDs) of n. Given 2n>6 which is not a Goldbug Number there exists a prime by Bertrandâ€™s Postulate that must be a PND of n or a Goldbach Pair. If 2n mod 4=0 then there exists n/20 then there exists (n+1)/2p mod d=0
This is a contradiction since p is prime.
Lemma 3
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Given p, a PND of n, 2n-p is not divisible by p.
2n-p mod p=0
->2n mod p=0