9846

Polya Conjecture

Define a number as of odd kind if the number of its prime factors is odd (taking multiplicity into account), and to be of even kind if the number of prime factors is even. Let and be the sum of the numbers of integers less than or equal to of the even and odd kinds.
Polya conjectured in 1919 that for all , . In 1962, Lehman found the first counterexample: 906150257. The first plot is of , which is closely related to the Liouville function . It shows fluctuations with increasing maxima alternating with minima near the axis. The second plot shows versus .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Reference
[1] Matt. "Examples of Apparent Patterns That Eventually Fail." Mathematics StackExchange. (Aug 27, 2014) math.stackexchange.com/questions/111440/examples-of-apparent-%5 Cpatterns-that-eventually-fail.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+