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Two conjectures of B. R. Santos concerning totitives


Authors: H. G. Kopetzky and W. Schwarz
Journal: Math. Comp. 33 (1979), 841-844
MSC: Primary 10A20; Secondary 10H25
DOI: https://doi.org/10.1090/S0025-5718-1979-0521300-8
MathSciNet review: 521300
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Abstract: Recently B. R. Santos conjectured that 12 is the largest integer n with the following property: $ (\ast)$

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{\text... ...ext{number}}{\text{.}}} \hfill \\ \end{array} } \hfill \\ \end{array} } \right.$

Using deep numerical estimates of Rosser and Schoenfeld for the number $ \pi (x)$ of primes less than x, it is proved that the conjecture of Santos is true. The same result holds, if in addition it is assumed in $ (\ast)$ that m is a prime.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1979-0521300-8
Keywords: Prime number theorem, prime totitives
Article copyright: © Copyright 1979 American Mathematical Society