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"

Questions:

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?

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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

Solution

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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