### 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 language | English (US) |
---|---|

Pages (from-to) | 230-235 |

Number of pages | 6 |

Journal | Mathematical Logic Quarterly |

Volume | 61 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2015 |

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### ASJC Scopus subject areas

- Logic

### Cite this

**Schatunowsky's theorem, Bonse's inequality, and Chebyshev's theorem in weak fragments of Peano arithmetic.** / Pambuccian, Victor.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Schatunowsky's theorem, Bonse's inequality, and Chebyshev's theorem in weak fragments of Peano arithmetic

AU - Pambuccian, Victor

PY - 2015/5/1

Y1 - 2015/5/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84929672688&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929672688&partnerID=8YFLogxK

U2 - 10.1002/malq.201400064

DO - 10.1002/malq.201400064

M3 - Article

AN - SCOPUS:84929672688

VL - 61

SP - 230

EP - 235

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

SN - 0942-5616

IS - 3

ER -