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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
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.

References [Enhancements On Off] (What's this?)

  • [1] E. LANDAU, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig und Berlin, 1909.
  • [2] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689
  • [3] J. B. ROSSER & L. SCHOENFELD, Sharper Bounds for the Chebyshev Functions $ \theta (x)$ and $ \psi (x)$, University of Wisconsin MRC Technical Summary Report #1475, 1974.
  • [4] Bernardo Recamán Santos, Twelve and its totitives, Math. Mag. 49 (1976), no. 5, 239–240. MR 0417037

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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