Euler pseudoprime

An Euler pseudoprimeMathworldPlanetmath p to a base b is a composite numberMathworldPlanetmath for which the congruenceMathworldPlanetmathPlanetmathPlanetmathPlanetmath


holds true, where (an) is the Jacobi symbolMathworldPlanetmath.

For example, given b=2, our Jacobi symbol (2p) 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 pseudoprimeMathworldPlanetmathPlanetmath ( 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.


  • 1 R. Crandall & C. Pomerance, Prime NumbersMathworldPlanetmath: A Computational Perspective, Springer, NY, 2001: 5.1
  • 2 B. Fine & G. Rosenberger, Number TheoryMathworldPlanetmathPlanetmath: An Introduction via the Distribution of the Primes Boston: Birkhäuser, 2007: Definition
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