On 2-powerfully perfect numbers in three quadratic rings

Colin Defant
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 2, Pages 1–11
Full paper (PDF, 208 Kb)

Details

Authors and affiliations

Colin Defant
Department of Mathematics, University of Florida
1400 Stadium Rd.
Gainesville, FL 32611
United States

Abstract

Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call n-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of 2-powerfully perfect numbers in the rings and the three imaginary quadratic rings with unique factorization in which 2 is not a prime.

Keywords

• Abundancy index
• Quadratic ring
• Solitary number
• Perfect number

AMS Classification

• Primary 11R11
• Secondary 11N80

References

1. Defant, C. (2014) An extension of the abundancy index to certain quadratic rings, Int. J. Math Comput. Sci. 9, 63–82.
2. Defant, C. (2014) Multiperfect numbers in certain quadratic rings, Int. J. Math Comput. Sci. 9, 49–61
3. Stark, H. M. (1967) A complete determination of the complex quadratic fields of classnumber one. Michigan Math. J. 14 1–27.