The generalized Schatunowsky theorem is provable in the weak arithmetic PA^- plus six elementary number-theoretic axioms, and one of those axioms is shown not to be needed.
Kaye, Models of Peano Arithmetic (Oxford University Press, 1991)
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Generalized Schatunowsky theorem in a weak arithmetic
The generalized Schatunowsky theorem is provable in the weak arithmetic PA^- plus six elementary number-theoretic axioms, and one of those axioms is shown not to be needed.