Schatunowsky's theorem, Bonse's inequality, and Chebyshev's theorem in weak fragments of Peano arithmetic

Abstract

In 1893, Schatunowsky showed that 30 is the largest number all of whose totatives are primes; we show that this result cannot be proved, in any form, in Chebyshev's theorem (Bertrand's postulate), even if all irreducibles are primes. Bonse's inequality is shown to be indeed weaker than Chebyshev's theorem. Schatunowsky's theorem holds in together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.