In the cafe today,
more insight was given as to how mathematics is an ongoing process. In order to
discover and learn, a mathematician needs to be constantly asking why, and how?
Indeed, we had previously proved that there are infinitely many primes. But the
question today was, now what? What other questions can we ask to further
understand prime numbers? Who cares that there are infinitely many primes? How
is this theorem useful in discovering new concepts and ideas?

Some questions asked by the group included: Is there a pattern among the distribution of prime numbers? Are there infinitely many primes of the form \(n^2 + 1\)? How many prime numbers are there less than n, for any integer n? This last question was actually conjectured by Gauss and Legendre, and a strong approximation is given in the famous Prime Number Theorem. We discussed this interesting result:

$$ \pi(x) \sim \frac{x}{\log x} $$

i.e. \(\lim \frac{\pi(x) }{\log x /x} = 1 \, \, x\to \infty \), where \( \pi (x) \) is the number of primes less than \( x \). The group spent the rest of the time proving Euler’s amazing Product Formula and one of its consequences. See the other post about Day 3.

This powerful formula gave a classic example as to why a mathematician, or anyone, should be constantly asking “now what?” Indeed the group then proved an amazing corollary to this formula: that \( \sum \frac{1}{p} = \infty \). This proposition seems very strange indeed.

Some questions asked by the group included: Is there a pattern among the distribution of prime numbers? Are there infinitely many primes of the form \(n^2 + 1\)? How many prime numbers are there less than n, for any integer n? This last question was actually conjectured by Gauss and Legendre, and a strong approximation is given in the famous Prime Number Theorem. We discussed this interesting result:

$$ \pi(x) \sim \frac{x}{\log x} $$

i.e. \(\lim \frac{\pi(x) }{\log x /x} = 1 \, \, x\to \infty \), where \( \pi (x) \) is the number of primes less than \( x \). The group spent the rest of the time proving Euler’s amazing Product Formula and one of its consequences. See the other post about Day 3.

This powerful formula gave a classic example as to why a mathematician, or anyone, should be constantly asking “now what?” Indeed the group then proved an amazing corollary to this formula: that \( \sum \frac{1}{p} = \infty \). This proposition seems very strange indeed.

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