TY - JOUR

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

AU - Pambuccian, Victor

N1 - Publisher Copyright:
© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

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

SN - 0942-5616

VL - 61

SP - 230

EP - 235

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

IS - 3

ER -