Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-21T01:27:31.511Z Has data issue: false hasContentIssue false

The morphology of ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $

Published online by Cambridge University Press:  21 October 2019

Andrew J. Simoson*
Affiliation:
King University, Bristol, Tennessee 37620, USA e-mail: ajsimoso@king.edu

Extract

What are the units, irreducibles, and primes of the ring ${\mathbb{Z}}\sqrt n $, the set of all numbers $a + b\sqrt n $ where a and b are integers and n is a fixed positive square-free integer? In the ring ${\mathbb{Z}}$, primes and irreducibles are synonymous and its units are ±1. ${\mathbb{Z}}\sqrt n $ is wilder, and our modest goal here is to catalogue all such numbers for ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $, where a and b range from 0 to 10; the result appears in Figure 1. Here are a few teasers that may induce a reader to read on: $3 + \sqrt {10} $ is a unit; 2, 3, 5, and 7 are irreducibles, but not 31; and 7 is the least positive integer that is prime in both ${\mathbb{Z}}$ and ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $.

Type
Articles
Copyright
© Mathematical Association 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barbeau, E. J., Pell’s equation, Springer Press (2003).10.1007/b97610CrossRefGoogle Scholar
Mollin, R. A., All solutions of the Diophantine equation x 2Dy 2 = n, Far East. J. Math. Sci., Special Vol. Part III (1998) pp. 257293.Google Scholar
Rosen, K. H., Elementary number theory and its applications, Addison Wesley Press, Boston (2011).Google Scholar
Dudley, U., A guide to elementary number theory, Mathematical Association of America (2009).Google Scholar