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

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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 PA-+ 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 PA- together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.

Original languageEnglish (US)
Pages (from-to)230-235
Number of pages6
JournalMathematical Logic Quarterly
Volume61
Issue number3
DOIs
StatePublished - May 1 2015

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ASJC Scopus subject areas

  • Logic

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