Problems & Puzzles: Conjectures

Conjecture 22. A stronger version of the Goldbach Conjecture

Mr. Rudolf Knjzek, from Austria, sent the following conjecture evidently related to the Goldbach Conjecture (GC):

For each even integer N>4 there is at least one prime number sqrt(N)<p<=N/2 so that q=N-p is also prime and N=p+q.

I will call this statement the Goldbach-Knjzek conjecture.

Knjzek says "To proof this will proof GC. And I think this will be not so difficult, than proofing the original conjecture". Later he added "My conjecture says that you need not the small primes to satisfy GC"


1. Would you like to try to proof the Goldbach-Knjzek conjecture?
2. Do you think - as
Knjzek - that to proof this conjecture will be not so difficult than proofing the original GC?

Note: The GC states that "Every even number 2.n=>4 is equal to the sum of two prime numbers". See our Conjecture 1  for more information about the GC


C. Rivera has narrowed the width of the range of the Goldbach-Knjzek  conjecture to sqrt(N)<p<4*sqrt(N), for N>4. He does not know if this is worthwhile.

He also notices that k*sqrt(N)<=N/2, for N=>4*k^2. Accordingly, the new range means a true narrower band-width for N=>64 while for the rest of the range 4<N<64, sqrt(N)<p<4*sqrt(N) is a wider band than the original one.





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