Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. B. ROSSER & L. SCHOENFELD, "Approximate formulas for some functions of prime numbers," Illinois J. Math., v. 6, 1962, pp. 64-94. MR 0137689 (25:1139)
  • [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] B. R. SANTOS, "Twelve and its totitives," Math. Mag., v. 49, 1976, pp. 239-240. MR 0417037 (54:5098)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10A20, 10H25

Retrieve articles in all journals with MSC: 10A20, 10H25

Additional Information

Keywords: Prime number theorem, prime totitives
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society