holds true, where is the Jacobi symbol.
For example, given , our Jacobi symbol with odd will be either 1 or . Then, for , 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 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.
|Date of creation||2013-03-22 16:49:48|
|Last modified on||2013-03-22 16:49:48|
|Last modified by||PrimeFan (13766)|