Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-30T17:05:14.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Carmichael Numbers with a Square Totient

Published online by Cambridge University Press:  20 November 2018

W. D. Banks
W. D. Banks*
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211 USA e-mail: bbanks@math.missouri.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\varphi$ denote the Euler function. In this paper, we show that for all large $x$ there are more than ${{x}^{0.33}}$ Carmichael numbers $n\,\le \,x$ with the property that $\varphi \left( n \right)$ is a perfect square. We also obtain similar results for higher powers.

Keywords

11N2511A25

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 1 , 01 March 2009 , pp. 3 - 8
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Alford, W. R., Granville, A., and Pomerance, C., There are infinitely many Carmichael numbers. Ann. of Math. (2) 139(1994), no. 3, 703–722.Google Scholar
[2] Baker, R. and Harman, G., Shifted primes without large prime factors. Acta Arith. 83(1998), no. 4, 331–361.Google Scholar
[3] Banks, W., Friedlander, J. B., Pomerance, C. and Shparlinski, I. E., Multiplicative structure of values of the Euler function. In: High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun. 41, American Mathematical Society, Providence, RI, 2004, pp. 29–47.Google Scholar
[4] Harman, G., On the number of Carmichael numbers up to x. Bull. LondonMath. Soc. 37(2005), no. 5, 641–650.Google Scholar
[5] Iwaniec, H., Almost-primes represented by quadratic polynomials. Invent. Math. 47(1978), no. 2, 171–188.Google Scholar