World Library  
Flag as Inappropriate
Email this Article

Pseudoprime

Article Id: WHEBN0028787420
Reproduction Date:

Title: Pseudoprime  
Author: World Heritage Encyclopedia
Language: English
Subject: Integer sequence, Fermat number, PSP (disambiguation), Frobenius pseudoprime
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Pseudoprime

A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy.

Some sources use the term pseudoprime about all probable primes, both composite numbers and actual primes.

Pseudoprimes are of primary importance in public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number.[1] However, finding and factoring the proper prime numbers for this use is correspondingly expensive, so various probabilistic primality tests are used to find primes amongst large numbers, some of which in rare cases incorrectly identify composite numbers as primes. On the other hand, deterministic primality tests, such as the AKS primality test, do not give false positives; there are no pseudoprimes with respect to them.

Fermat pseudoprimes

Main article: Fermat pseudoprime

Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. If a composite integer x is coprime to an integer a > 1 and x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a. Some sources use variations of this definition, for example to only allow odd numbers to be pseudoprimes.[2]

An integer x that is a Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number.

Classes

References

nl:Kleine stelling van Fermat#Pseudo-priemgetallen
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.