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Goldbach conjecture sequences in quantum mechanics

We show that there is a correspondence between Goldbach conjecture sequences (GCS) and expectation values of the number operator in Fock states. We demonstrate that depending on the normalization or not of Fock state superpositions, we have sequences that are equivalent and sequences that are not equivalent to GCS. We propose an algorithm where sequences equivalent to GCS can be derived in terms of expectation values with normalized states. Defining states whose projections generate GCS, we relate this problem to eigenstates of quantum harmonic oscillator and discuss Fock states directly associated to GCS, taking into account the hamiltonian spectrum and quantum vacuum fluctuations. Finally, we address the problems of degeneracy, maps associating GCS and Goldbach partitions.