From the Fun Fact files, here is a Random Fun Fact, at the Advanced level: Euler's Product Formula
Here is an amazing formula due to Euler:
SUM_{n=1 to infinity} n^{s}
= PROD_{p prime} (1  p^{s})^{1} .
What's interesting about this formula is that it
relates an expression involving all the positive integers
to one involving just primes!
And you can use it to prove there must be infinitely many primes.
For, if there were only finitely many primes, then the right side of the expression is a finite product, and in particular for s=1. But for s=1, the left side of the equation is the
harmonic series which we know must diverge!
This is a contradiction, so there must be infinitely many primes.
Presentation Suggestions:
Interested students may wish to take a few terms on the
right hand side, use a power expansion, and
multiply them out... to get an idea of why the equality
holds.
The Math Behind the Fact:
The left hand side, when s is viewed as a complex variable,
is also known as the Riemann zeta function.
Because of the above relationship, the study of
zeta functions is closely related to the study
of the distribution of primes!
How to Cite this Page:
Su, Francis E., et al. "Euler's Product Formula."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
The Link for this Fun Fact:
is directly accessible here.
