Euler pseudoprime
An Euler pseudoprime^{} $p$ to a base $b$ is a composite number^{} for which the congruence^{}
$${b}^{\frac{p-1}{2}}\equiv \left(\frac{b}{p}\right)modp$$ |
holds true, where $\left(\frac{a}{n}\right)$ is the Jacobi symbol^{}.
For example, given $b=2$, our Jacobi symbol $\left(\frac{2}{p}\right)$ with $p$ odd will be either 1 or $-1$. Then, for $p=561$, the Jacobi symbol is 1. Next, we see that 2 raised to the 280th is 1942668892225729070919461906823518906642406839052139521251812409738904285205208498176, which is one more than 561 times 3462867900580622229802962400754935662464183313818430519165441015577369492344400175. Hence 561 is an Euler pseudoprime. The next few Euler pseudoprimes to base 2 are 1105, 1729, 1905, 2047, 2465, 4033, 4681 (see A047713 in Sloane’s OEIS). An Euler pseudoprime is sometimes called an Euler-Jacobi pseudoprime, to distinguish it from a pseudoprime^{} (http://planetmath.org/PseudoprimeP) for which the congruence can be either to 1 or $-1$ regardless of the Jacobi symbol (341 is then an Euler pseudoprime under this relaxed definition). Both terms are also sometimes used alone with 2 as the implied base.
If a composite number is an Euler pseudoprime to a given base, it is also a regular pseudoprime to that base, but not all regular pseudoprimes to that base are also Euler pseudoprimes to it.
References
- 1 R. Crandall & C. Pomerance, Prime Numbers^{}: A Computational Perspective, Springer, NY, 2001: 5.1
- 2 B. Fine & G. Rosenberger, Number Theory^{}: An Introduction via the Distribution of the Primes Boston: Birkhäuser, 2007: Definition 5.3.1.4
Title | Euler pseudoprime |
---|---|
Canonical name | EulerPseudoprime |
Date of creation | 2013-03-22 16:49:48 |
Last modified on | 2013-03-22 16:49:48 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 5 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 11A51 |
Synonym | Euler-Jacobi pseudoprime |