Polya Conjecture

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.

[more]

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 .

[less]

Contributed by: Enrique Zeleny (August 2014)
Open content licensed under CC BY-NC-SA


Snapshots


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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send