Large Counterexample

A positive integer is said to be of even type if its factorization into primes has an even number of primes. Otherwise it is said to be of odd type. Examples: 4=2*2 is even type, 18=2*3*3 is odd type. (We say 1 has 0 primes and is therefore of even type.)

Let E(n)= the number of positive integers <= of even type. 
Let O(n)= the number of positive integers <= n of odd type. 
What can be said about the relative size of E(n) and O(n)? Are there more of one than the other?

Perhaps O(n) >= E(n) for all n>=2? After all, products of primes come “before” products of two primes…

This statement is known as Polya’s conjecture, and dates back from 1919. After it was checked for all n <= a million, many people believed it had to be true. But a belief is not a proof… and in fact the conjecture is false!

In 1962, Lehman found a counterexample: at n=906180359, it is the case that O(n)=E(n)-1.

Presentation Suggestions:
Students may be able to come up with a conjecture if you start with some examples. You may wish to make the conjecture more plausible with some other “heuristic” arguments.

The Math Behind the Fact:
This example drives home the point that “obvious” facts, checked for many cases, to not constitute a proof for all integers!

How to Cite this Page: 
Su, Francis E., et al. “Large Counterexample.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
H. Stark, An Introduction to Number Theory, MIT Press, 1987.

Fun Fact suggested by:
Lesley Ward

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