World Library  
Flag as Inappropriate
Email this Article

Carol number

Article Id: WHEBN0003422372
Reproduction Date:

Title: Carol number  
Author: World Heritage Encyclopedia
Language: English
Subject: Proth number, Lucas number, Double Mersenne number, Thabit number, Million
Collection: Integer Sequences
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Carol number

A Carol number is an integer of the form 4^n - 2^{n + 1} - 1. An equivalent formula is (2^n - 1)^2 - 2. The first few Carol numbers are: −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527 (sequence A093112 in OEIS).

Carol numbers were first studied by Cletus Emmanuel, who named them after a friend, Carol G. Kirnon.[1][2]

Contents

  • Binary representation 1
  • Primes and modular relations 2
  • References 3
  • External links 4

Binary representation

For n > 2, the binary representation of the n-th Carol number is n − 2 consecutive ones, a single zero in the middle, and n + 1 more consecutive ones, or to put it algebraically,

\sum_{i \ne n + 2}^{2n} 2^{i - 1}.

So, for example, 47 is 101111 in binary, 223 is 11011111, etc. The difference between the 2n-th Mersenne number and the n-th Carol number is 2^{n + 1}. This gives yet another equivalent expression for Carol numbers, (2^{2n} - 1) - 2^{n + 1}. The difference between the n-th Kynea number and the n-th Carol number is the (n + 2)th power of two.

Primes and modular relations

Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a prime number, its index n cannot be of the form 3x + 2 for x > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127 (these are listed in Sloane's  A091516). As of July 2007, the largest known Carol number that is also a prime is the Carol number for n = 253987, which has 152916 digits.[3][4] It was found by Cletus Emmanuel in May 2007, using the programs MultiSieve and PrimeFormGW. It is the 40th Carol prime.

The 7th Carol number and 5th Carol prime, 16127, is also a prime when its digits are reversed, so it is the smallest Carol emirp.[5] The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.[6]

References

  1. ^ Cletus Emmanuel at Prime Pages
  2. ^ Message to Yahoo primenumbers group from Cletus Emmanuel
  3. ^ Entry for 253987th Carol number at Prime Pages
  4. ^ Carol Primes and Kynea Primes by Steven Harvey
  5. ^ Prime Curios 16127 at Prime Pages
  6. ^ Prime Curios 16769023 at Prime Pages

External links

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.