Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Medium level:

Sum of Prime Reciprocals

It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges.

But what about the sum of reciprocals of the prime numbers?

These diverge, too!

One way to interpret this fact is that there must be a "lot" of primes---well, of course there are an infinite number of them, but not every infinite set of natural numbers has a reciprocal sum which diverges (for instance, take the powers of 2). So, while primes get sparser and sparser the farther you go out, they are not as sparse as the powers of 2.

Presentation Suggestions:
This is best done after you have shown in class that the harmonic series diverges.

The Math Behind the Fact:
Euler first noted this fact, and one proof can be obtained by taking the natural logarithm of both sides of Euler's Product Formula, (using s=1 in that formula) and noting that the right hand side consists of terms of the form

Log(p/p-1) = Log(1 + (1/p-1)),

where Log denotes the natural log, and p is a prime. Using a Taylor series for Log, this term is itself bounded by 1/(p-1) < 1/p. Thus, if the sum of reciprocals for primes converge, then the harmonic series would converge, a contradiction.

There are many refined questions you can ask about the number of primes. See the Fun Fact How Many Primes.

How to Cite this Page:
Su, Francis E., et al. "Sum of Prime Reciprocals." Math Fun Facts. <>.

Keywords:    number theory
Subjects:    number theory
Level:    Medium
Fun Fact suggested by:   Lesley Ward
Suggestions? Use this form.
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!