# Proving theorems and special cases (Part 3): Skewes’ number

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no… even checking many special cases of a conjecture does not mean that the conjecture is correct.

In the first two posts of this series, we showed conjectures could be true for the first 40 cases or even the first 900 million odd cases but fail on the next case.  In today’s post, I’ll describe a conjecture that, for hundreds of years, was thought to be true for all integers $n$ and has since been shown to be true for all $n \le 10^{316}$. However, despite being true for so many special cases, the conjecture is false.

Let’s absorb the above paragraph again. The conjecture “$n^2 - n + 41$ is always prime” was true for the first 40 cases before failing. By contrast, the conjecture I’m about the describe is true for the first (approximately) 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000 cases before failing.

The conjecture in question concerns the number of prime numbers $\pi(n)$ that are less than or equal to $n$. For example:

There are four prime numbers less than 10 (namely 2, 3, 5, 7), and so $\pi(10) = 4$.

There are eight prime numbers less than 10 (namely 2, 3, 5, 7, 11, 13, 17, 19), and so $\pi(20) = 8$.

One of the classic problems in mathematics, often called the Prime Number Theorem, is estimating the value of $\pi(n)$ when $n$ is large. It turns out that one way of approximating $\pi(n)$ is through the logarithmic integral

$\hbox{Li}(n) = \displaystyle \int_2^n \frac{dx}{\ln x}$

Indeed, the Prime Number Theorem states that

$\pi(n) \sim \hbox{Li}(n)$,

or

$\displaystyle \lim_{n \to \infty} \frac{\pi(n)}{\hbox{Li}(n)} = 1$.

This asymptotic relationship was proven by the eminent mathematician Karl Friedrich Gauss.

With that as prelude, here’s the false conjecture. For decades, luminaries of mathematics, including Gauss and Bernhard Riemann, conjectured that

$\pi(n) < \hbox{Li}(n)$ for all integers $n$

based on the available numerical evidence at the time. However, this conjecture was proven false in the 20th century by the British mathematician John Littlewood. No only did Littlewood show that there is a value of $n$ so that $\pi(n) > \hbox{Li}(n)$, but he also showed that the graphs of $\pi(n)$ and $\hbox{Li}(n)$ cross over each other infinitely many times!

Littlewood proved his theorem over 100 years ago in 1914. Today, even with modern computing power, we still do not precisely know the first value of $n$ for which $\pi(n) > \hbox{Li}(n)$. This number is called Skewes’ number, in honor of the mathematician who first found (in 1955) an upper bound on this first crossing point. The bound that he found was absolutely enormous: he showed that $\pi(n) > \hbox{Li}(n)$ for some $n$ less than

$\displaystyle 10^{\displaystyle 10^{10^{34}}}$

More recent work has established that the first crossover likely occurs in the vicinity of $n \approx 1.397 \times 10^{316}$.

Whoever first evaluates Skewes’ number exactly will surely have a nice feather in his/her cap for completing a task that mystified both Gauss and Riemann.

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