A Concrete Introduction to Higher Algebra pp 413-431 | Cite as

# Carmichael Numbers

Chapter

This chapter returns to the question of deciding whether a given odd number *m* is prime. The *a*-pseudoprime test of Chapter 10B will not work on Carmichael numbers. We first describe an idea of Alford that shows that there are many Carmichael numbers. Then we develop the strong *a*-pseudoprime test and prove that every composite number *m* fails the strong *a*-pseudoprime test for at least half of the numbers *a* < *m.* Thus there are no composite numbers that are “strong Carmichael numbers”.

## Keywords

Prime Divisor Chinese Remainder Theorem Fermat Number Composite Number Generalize Riemann Hypothesis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer-Verlag Berlin Heidelberg 2009